:: Definitions and Basic Properties of Boolean & Union of Many
:: Sorted Sets
:: by Artur Korni{\l}owicz
::
:: Received April 27, 1995
:: Copyright (c) 1995-2017 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies PBOOLE, ZFMISC_1, FUNCT_1, RELAT_1, XBOOLE_0, FUNCT_4, FUNCOP_1,
TARSKI, FUNCT_2, FINSET_1, SETLIM_2;
notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, FUNCT_1, FUNCT_2, FINSET_1,
FUNCT_4, FUNCOP_1, PBOOLE;
constructors FUNCT_4, PBOOLE, FINSET_1, RELSET_1;
registrations XBOOLE_0, SUBSET_1, FUNCOP_1, PRE_CIRC;
requirements SUBSET, BOOLE;
definitions PBOOLE;
equalities FUNCOP_1;
expansions PBOOLE;
theorems CARD_2, SETFAM_1, FUNCOP_1, FRAENKEL, FUNCT_4, PBOOLE, FINSET_1,
TARSKI, ZFMISC_1, XBOOLE_0, XBOOLE_1, PARTFUN1, RELAT_1, FUNCT_1;
schemes PBOOLE;
begin :: Boolean of Many Sorted Sets
reserve x, y for object, I for set,
A, B, X, Y for ManySortedSet of I;
definition
let I, A;
func bool A -> ManySortedSet of I means
:Def1:
for i be object st i in I holds it.i = bool (A.i);
existence
proof
deffunc V(object) = bool (A.$1);
thus ex X being ManySortedSet of I st
for i be object st i in I holds X.i = V(i) from PBOOLE:sch 4;
end;
uniqueness
proof
let X, Y be ManySortedSet of I such that
A1: for i be object st i in I holds X.i = bool (A.i) and
A2: for i be object st i in I holds Y.i = bool (A.i);
now
let i be object;
assume
A3: i in I;
hence X.i = bool (A.i) by A1
.= Y.i by A2,A3;
end;
hence X = Y;
end;
end;
registration
let I, A;
cluster bool A -> non-empty;
coherence
proof
let i be object such that
A1: i in I;
bool (A.i) is non empty;
hence thesis by A1,Def1;
end;
end;
Lm1: for i, I, X be set for M be ManySortedSet of I st i in I holds dom (M +*
(i .--> X)) = I
proof
let i, I, X be set, M be ManySortedSet of I such that
A1: i in I;
thus dom (M +* (i .--> X)) = dom M \/ dom (i .--> X) by FUNCT_4:def 1
.= I \/ dom (i .--> X) by PARTFUN1:def 2
.= I \/ {i} by FUNCOP_1:13
.= I by A1,ZFMISC_1:40;
end;
Lm2: for i be set st i in I holds (bool (A (\/) B)).i = bool (A.i \/ B.i)
proof
let i be set;
assume
A1: i in I;
hence (bool (A (\/) B)).i = bool ((A (\/) B).i) by Def1
.= bool (A.i \/ B.i) by A1,PBOOLE:def 4;
end;
Lm3: for i be set st i in I holds (bool (A (/\) B)).i = bool (A.i /\ B.i)
proof
let i be set;
assume
A1: i in I;
hence (bool (A (/\) B)).i = bool ((A (/\) B).i) by Def1
.= bool (A.i /\ B.i) by A1,PBOOLE:def 5;
end;
Lm4: for i be set st i in I holds (bool (A (\) B)).i = bool (A.i \ B.i)
proof
let i be set;
assume
A1: i in I;
hence (bool (A (\) B)).i = bool ((A (\) B).i) by Def1
.= bool (A.i \ B.i) by A1,PBOOLE:def 6;
end;
Lm5: for i be set st i in I holds (bool (A (\+\) B)).i = bool (A.i \+\ B.i)
proof
let i be set;
assume
A1: i in I;
hence (bool (A (\+\) B)).i = bool ((A (\+\) B).i) by Def1
.= bool (A.i \+\ B.i) by A1,PBOOLE:4;
end;
theorem Th1: :: Tarski:6
X = bool Y iff for A holds A in X iff A c= Y
proof
thus X = bool Y implies for A holds A in X iff A c= Y
proof
assume
A1: X = bool Y;
let A;
thus A in X implies A c= Y
proof
assume
A2: A in X;
let i be object;
assume
A3: i in I;
then
A4: A.i in X.i by A2;
X.i = bool (Y.i) by A1,A3,Def1;
hence thesis by A4;
end;
assume
A5: A c= Y;
let i be object;
assume
A6: i in I;
then
A7: A.i c= Y.i by A5;
X.i = bool (Y.i) by A1,A6,Def1;
hence thesis by A7;
end;
assume
A8: for A holds A in X iff A c= Y;
now
let i be object such that
A9: i in I;
EmptyMS I c= Y by PBOOLE:43;
then
A10: EmptyMS I in X by A8;
for A9 be set holds A9 in X.i iff A9 c= Y.i
proof
let A9 be set;
A11: dom (i .--> A9) = {i} by FUNCOP_1:13;
dom (EmptyMS I +* (i .--> A9)) = I by A9,Lm1;
then reconsider K = EmptyMS I +* (i .--> A9) as ManySortedSet of I by
PARTFUN1:def 2,RELAT_1:def 18;
i in {i} by TARSKI:def 1;
then
A12: K.i = (i .--> A9).i by A11,FUNCT_4:13
.= A9 by FUNCOP_1:72;
thus A9 in X.i implies A9 c= Y.i
proof
assume
A13: A9 in X.i;
K in X
proof
let j be object such that
A14: j in I;
now
per cases;
case
j = i;
hence thesis by A12,A13;
end;
case
j <> i;
then not j in dom (i .--> A9) by A11,TARSKI:def 1;
then K.j = EmptyMS I.j by FUNCT_4:11;
hence thesis by A10,A14;
end;
end;
hence thesis;
end;
then K c= Y by A8;
hence thesis by A9,A12;
end;
assume
A15: A9 c= Y.i;
K c= Y
proof
let j be object such that
A16: j in I;
now
per cases;
case
j = i;
hence thesis by A12,A15;
end;
case
j <> i;
then not j in dom (i .--> A9) by A11,TARSKI:def 1;
then
A17: K.j = EmptyMS I.j by FUNCT_4:11;
EmptyMS I c= Y by PBOOLE:43;
hence thesis by A16,A17;
end;
end;
hence thesis;
end;
then K in X by A8;
hence thesis by A9,A12;
end;
then X.i = bool (Y.i) by ZFMISC_1:def 1;
hence X.i = (bool Y).i by A9,Def1;
end;
hence thesis;
end;
theorem :: ZFMISC_1:1
bool EmptyMS I = I --> {{}}
proof
now
let i be object;
assume
A1: i in I;
then (bool EmptyMS I).i = bool (EmptyMS I.i) by Def1
.= bool{} by PBOOLE:5
.= {{}} by ZFMISC_1:1;
hence (bool EmptyMS I).i = (I --> {{}}).i by A1,FUNCOP_1:7;
end;
hence thesis;
end;
theorem
for x being set holds
bool (I --> x) = I --> bool x
proof let x be set;
now
let i be object;
assume
A1: i in I;
hence (bool (I --> x)).i = bool ((I --> x).i) by Def1
.= bool x by A1,FUNCOP_1:7
.= (I --> bool x).i by A1,FUNCOP_1:7;
end;
hence thesis;
end;
theorem :: ZFMISC_1:30
bool (I --> {x}) = I --> { {} , {x} }
proof
now
let i be object;
assume
A1: i in I;
hence (bool (I --> {x})).i = bool ((I --> {x}).i) by Def1
.= bool {x} by A1,FUNCOP_1:7
.= { {} , {x} } by ZFMISC_1:24
.= (I --> { {} , {x} }).i by A1,FUNCOP_1:7;
end;
hence thesis;
end;
theorem :: ZFMISC_1:76
EmptyMS I c= A
proof
let i be object;
assume i in I;
EmptyMS I.i = {} by PBOOLE:5;
hence thesis by XBOOLE_1:2;
end;
theorem :: ZFMISC_1:79
A c= B implies bool A c= bool B
proof
assume
A1: A c= B;
let i be object;
assume
A2: i in I;
then
A3: A.i c= B.i by A1;
(bool A).i = bool (A.i) & (bool B).i = bool (B.i) by A2,Def1;
hence thesis by A3,ZFMISC_1:67;
end;
theorem :: ZFMISC_1:81
bool A (\/) bool B c= bool (A (\/) B)
proof
let i be object;
assume
A1: i in I;
then
A2: (bool (A (\/) B)).i = bool (A.i \/ B.i) by Lm2;
(bool A (\/) bool B).i = (bool A).i \/ (bool B).i by A1,PBOOLE:def 4
.= bool (A.i) \/ (bool B).i by A1,Def1
.= bool (A.i) \/ bool (B.i) by A1,Def1;
hence thesis by A2,ZFMISC_1:69;
end;
theorem :: ZFMISC_1:82
bool A (\/) bool B = bool (A (\/) B) implies
for i be set st i in I holds A.i,B.i are_c=-comparable
proof
assume
A1: bool A (\/) bool B = bool (A (\/) B);
let i be set such that
A2: i in I;
bool (A.i \/ B.i) = (bool A (\/) bool B).i by A1,A2,Lm2
.= (bool A).i \/ (bool B).i by A2,PBOOLE:def 4
.= (bool A).i \/ (bool (B.i)) by A2,Def1
.= (bool (A.i)) \/ (bool (B.i)) by A2,Def1;
hence thesis by ZFMISC_1:70;
end;
theorem :: ZFMISC_1:83
bool (A (/\) B) = bool A (/\) bool B
proof
now
let i be object;
assume
A1: i in I;
hence bool (A (/\) B).i = bool (A.i /\ B.i) by Lm3
.= (bool (A.i)) /\ (bool (B.i)) by ZFMISC_1:71
.= (bool (A.i)) /\ (bool B.i) by A1,Def1
.= (bool A).i /\ (bool B.i) by A1,Def1
.= (bool A (/\) bool B).i by A1,PBOOLE:def 5;
end;
hence thesis;
end;
theorem :: ZFMISC_1:84
bool (A (\) B) c= (I --> {{}}) (\/) (bool A (\) bool B)
proof
let i be object;
assume
A1: i in I;
then
A2: (bool (A (\) B)).i = bool (A.i \ B.i) by Lm4;
((I --> {{}}) (\/) (bool A (\) bool B)).i
= (I --> {{}}).i \/ (bool A (\) bool B).i by A1,PBOOLE:def 4
.= {{}} \/ (bool A (\) bool B).i by A1,FUNCOP_1:7
.= {{}} \/ ((bool A).i \ (bool B).i) by A1,PBOOLE:def 6
.= {{}} \/ ((bool (A.i)) \ (bool B).i) by A1,Def1
.= {{}} \/ (bool (A.i) \ bool (B.i)) by A1,Def1;
hence thesis by A2,ZFMISC_1:72;
end;
theorem
X c= A (\) B iff X c= A & X misses B
proof
thus X c= A (\) B implies X c= A & X misses B
proof
assume X c= A (\) B;
then
A1: X in bool (A (\) B) by Th1;
thus X c= A
proof
let i be object;
assume
A2: i in I;
then X.i in (bool (A (\) B)).i by A1;
then X.i in bool (A.i \ B.i) by A2,Lm4;
hence thesis by XBOOLE_1:106;
end;
let i be object;
assume
A3: i in I;
then X.i in (bool (A (\) B)).i by A1;
then X.i in bool (A.i \ B.i) by A3,Lm4;
hence thesis by XBOOLE_1:106;
end;
assume
A4: X c= A & X misses B;
let i be object;
assume
A5: i in I;
then X.i c= A.i & X.i misses B.i by A4;
then X.i c= A.i \ B.i by XBOOLE_1:86;
hence thesis by A5,PBOOLE:def 6;
end;
theorem :: ZFMISC_1:86
bool (A (\) B) (\/) bool (B (\) A) c= bool (A (\+\) B)
proof
let i be object;
assume
A1: i in I;
then
A2: bool (A (\+\) B).i = bool (A.i \+\ B.i) by Lm5;
(bool (A (\) B) (\/) bool (B (\) A)).i
= (bool (A (\) B)).i \/ (bool (B (\) A) ).i
by A1,PBOOLE:def 4
.= (bool (A.i \ B.i)) \/ (bool (B (\) A)).i by A1,Lm4
.= (bool (A.i \ B.i)) \/ (bool (B.i \ A.i)) by A1,Lm4;
hence thesis by A2,ZFMISC_1:73;
end;
theorem :: ZFMISC_1:87
X c= A (\+\) B iff X c= A (\/) B & X misses A (/\) B
proof
thus X c= A (\+\) B implies X c= A (\/) B & X misses A (/\) B
proof
assume X c= A (\+\) B;
then
A1: X in bool (A (\+\) B) by Th1;
thus X c= A (\/) B
proof
let i be object;
assume
A2: i in I;
then X.i in (bool (A (\+\) B)).i by A1;
then X.i in bool (A.i \+\ B.i) by A2,Lm5;
then X.i c= A.i \/ B.i by XBOOLE_1:107;
hence thesis by A2,PBOOLE:def 4;
end;
let i be object;
assume
A3: i in I;
then X.i in (bool (A (\+\) B)).i by A1;
then X.i in bool (A.i \+\ B.i) by A3,Lm5;
then X.i misses A.i /\ B.i by XBOOLE_1:107;
hence thesis by A3,PBOOLE:def 5;
end;
assume that
A4: X c= A (\/) B and
A5: X misses A (/\) B;
let i be object;
assume
A6: i in I;
then X.i misses (A (/\) B).i by A5;
then
A7: X.i misses A.i /\ B.i by A6,PBOOLE:def 5;
X.i c= (A (\/) B).i by A4,A6;
then X.i c= A.i \/ B.i by A6,PBOOLE:def 4;
then X.i c= A.i \+\ B.i by A7,XBOOLE_1:107;
hence thesis by A6,PBOOLE:4;
end;
theorem :: ZFMISC_1:89
X c= A or Y c= A implies X (/\) Y c= A
proof
assume
A1: X c= A or Y c= A;
per cases by A1;
suppose
A2: X c= A;
let i be object;
assume
A3: i in I;
then X.i c= A.i by A2;
then X.i /\ Y.i c= A.i by XBOOLE_1:108;
hence thesis by A3,PBOOLE:def 5;
end;
suppose
A4: Y c= A;
let i be object;
assume
A5: i in I;
then Y.i c= A.i by A4;
then X.i /\ Y.i c= A.i by XBOOLE_1:108;
hence thesis by A5,PBOOLE:def 5;
end;
end;
theorem :: ZFMISC_1:90
X c= A implies X (\) Y c= A
proof
assume
A1: X c= A;
let i be object;
assume
A2: i in I;
then X.i c= A.i by A1;
then X.i \ Y.i c= A.i by XBOOLE_1:109;
hence thesis by A2,PBOOLE:def 6;
end;
theorem :: ZFMISC_1:91
X c= A & Y c= A implies X (\+\) Y c= A
proof
assume
A1: X c= A & Y c= A;
let i be object;
assume
A2: i in I;
then X.i c= A.i & Y.i c= A.i by A1;
then X.i \+\ Y.i c= A.i by XBOOLE_1:110;
then X.i \+\ Y.i in bool (A.i);
then (X (\+\) Y).i in bool (A.i) by A2,PBOOLE:4;
hence thesis;
end;
theorem :: ZFMISC_1:105
[|X, Y|] c= bool bool (X (\/) Y)
proof
let i be object;
assume
A1: i in I;
then
A2: [|X, Y|].i = [:X.i, Y.i:] by PBOOLE:def 16;
(bool bool (X (\/) Y)).i = bool ((bool (X (\/) Y)).i) by A1,Def1
.= bool bool (X.i \/ Y.i) by A1,Lm2;
hence thesis by A2,ZFMISC_1:86;
end;
theorem :: FIN_TOPO:4
X c= A iff X in bool A
proof
thus X c= A implies X in bool A
proof
assume
A1: X c= A;
let i be object;
assume
A2: i in I;
then X.i c= A.i by A1;
then X.i in bool (A.i);
hence thesis by A2,Def1;
end;
assume
A3: X in bool A;
let i be object;
assume
A4: i in I;
then X.i in (bool A).i by A3;
then X.i in bool (A.i) by A4,Def1;
hence thesis;
end;
theorem :: FRAENKEL:5
(Funcs)(A,B) c= bool [|A, B|]
proof
let i be object;
assume
A1: i in I;
then
A2: ((Funcs)(A,B)).i = Funcs (A.i, B.i) by PBOOLE:def 17;
(bool [|A, B|]).i = bool ([|A, B|].i) by A1,Def1
.= bool [:A.i, B.i:] by A1,PBOOLE:def 16;
hence thesis by A2,FRAENKEL:2;
end;
begin :: Union of Many Sorted Sets
definition
let I, A;
func union A -> ManySortedSet of I means
:Def2:
for i be object st i in I holds it.i = union (A.i);
existence
proof
deffunc V(object) = union (A.$1);
thus ex X being ManySortedSet of I st
for i be object st i in I holds X.i = V
(i) from PBOOLE:sch 4;
end;
uniqueness
proof
let X, Y be ManySortedSet of I such that
A1: for i be object st i in I holds X.i = union (A.i) and
A2: for i be object st i in I holds Y.i = union (A.i);
now
let i be object;
assume
A3: i in I;
hence X.i = union (A.i) by A1
.= Y.i by A2,A3;
end;
hence X = Y;
end;
end;
registration
let I;
cluster union EmptyMS I -> empty-yielding;
coherence
proof
let i be object;
A1: union (EmptyMS I.i) is empty by PBOOLE:5,ZFMISC_1:2;
assume i in I;
hence thesis by A1,Def2;
end;
end;
Lm6: for i be set st i in I holds (union (A (\/) B)).i = union (A.i \/ B.i)
proof
let i be set;
assume
A1: i in I;
hence (union (A (\/) B)).i = union ((A (\/) B).i) by Def2
.= union (A.i \/ B.i) by A1,PBOOLE:def 4;
end;
Lm7: for i be set st i in I holds (union (A (/\) B)).i = union (A.i /\ B.i)
proof
let i be set;
assume
A1: i in I;
hence (union (A (/\) B)).i = union ((A (/\) B).i) by Def2
.= union (A.i /\ B.i) by A1,PBOOLE:def 5;
end;
theorem :: Tarski:def 4
A in union X iff ex Y st A in Y & Y in X
proof
thus A in union X implies ex Y st A in Y & Y in X
proof
defpred P[object,object] means
ex B being set st B = $2 & A.$1 in B & $2 in X.$1;
assume
A1: A in union X;
A2: for i being object st i in I ex Y being object st P[i,Y]
proof
let i be object;
assume
A3: i in I;
then A.i in (union X).i by A1;
then A.i in union (X.i) by A3,Def2;
then consider B being set such that
A4: A.i in B & B in X.i by TARSKI:def 4;
take B;
thus thesis by A4;
end;
consider K be ManySortedSet of I such that
A5: for i be object st i in I holds P[i,K.i] from PBOOLE:sch 3(A2);
take K;
thus A in K
proof
let i be object;
assume i in I;
then P[i,K.i] by A5;
hence thesis;
end;
thus K in X
proof
let i be object;
assume i in I;
then P[i,K.i] by A5;
hence thesis;
end;
end;
given Y such that
A6: A in Y & Y in X;
let i be object;
assume
A7: i in I;
then A.i in Y.i & Y.i in X.i by A6;
then A.i in union (X.i) by TARSKI:def 4;
hence thesis by A7,Def2;
end;
theorem :: ZFMISC_1:2
union EmptyMS I = EmptyMS I
proof
now
let i be object;
assume i in I;
hence (union EmptyMS I).i = union (EmptyMS I.i) by Def2
.= {} by PBOOLE:5,ZFMISC_1:2
.= EmptyMS I.i by PBOOLE:5;
end;
hence thesis;
end;
theorem
for x being set holds
union (I --> x) = I --> union x
proof let x be set;
now
let i be object;
assume
A1: i in I;
hence (union (I --> x)).i = union ((I --> x).i) by Def2
.= union x by A1,FUNCOP_1:7
.= (I --> union x).i by A1,FUNCOP_1:7;
end;
hence thesis;
end;
theorem :: ZFMISC_1:31
union (I --> {x}) = I --> x
proof
now
let i be object;
assume
A1: i in I;
hence (union (I --> {x})).i = union ((I --> {x}).i) by Def2
.= union {x} by A1,FUNCOP_1:7
.= x by ZFMISC_1:25
.= (I --> x).i by A1,FUNCOP_1:7;
end;
hence thesis;
end;
theorem :: ZFMISC_1:32
union (I --> { {x},{y} }) = I --> {x,y}
proof
now
let i be object;
assume
A1: i in I;
hence (union (I --> {{x},{y}})).i = union ((I --> {{x},{y}}).i) by Def2
.= union {{x},{y}} by A1,FUNCOP_1:7
.= {x,y} by ZFMISC_1:26
.= (I --> {x,y}).i by A1,FUNCOP_1:7;
end;
hence thesis;
end;
theorem :: ZFMISC_1:92
X in A implies X c= union A
proof
assume
A1: X in A;
let i be object;
assume
A2: i in I;
then X.i in A.i by A1;
then X.i c= union (A.i) by ZFMISC_1:74;
hence thesis by A2,Def2;
end;
theorem :: ZFMISC_1:95
A c= B implies union A c= union B
proof
assume
A1: A c= B;
let i be object;
assume
A2: i in I;
then A.i c= B.i by A1;
then union (A.i) c= union (B.i) by ZFMISC_1:77;
then (union A).i c= union (B.i) by A2,Def2;
hence thesis by A2,Def2;
end;
theorem :: ZFMISC_1:96
union (A (\/) B) = union A (\/) union B
proof
now
let i be object;
assume
A1: i in I;
hence (union (A (\/) B)).i = union (A.i \/ B.i) by Lm6
.= union (A.i) \/ union (B.i) by ZFMISC_1:78
.= (union A).i \/ union (B.i) by A1,Def2
.= (union A).i \/ (union B).i by A1,Def2
.= (union A (\/) union B).i by A1,PBOOLE:def 4;
end;
hence thesis;
end;
theorem :: ZFMISC_1:97
union (A (/\) B) c= union A (/\) union B
proof
let i be object;
assume
A1: i in I;
then
A2: (union (A (/\) B)).i = union (A.i /\ B.i) by Lm7;
(union A (/\) union B).i = (union A).i /\ (union B).i by A1,PBOOLE:def 5
.= union (A.i) /\ (union B).i by A1,Def2
.= union (A.i) /\ union (B.i) by A1,Def2;
hence thesis by A2,ZFMISC_1:79;
end;
theorem :: ZFMISC_1:99
union bool A = A
proof
now
let i be object;
assume
A1: i in I;
hence (union bool A).i = union ((bool A).i) by Def2
.= union bool (A.i) by A1,Def1
.= A.i by ZFMISC_1:81;
end;
hence thesis;
end;
theorem :: ZFMISC_1:100
A c= bool union A
proof
let i be object;
assume
A1: i in I;
then (bool union A).i = bool ((union A).i) by Def1
.= bool union (A.i) by A1,Def2;
hence thesis by ZFMISC_1:82;
end;
theorem :: LATTICE4:1
union Y c= A & X in Y implies X c= A
proof
assume that
A1: union Y c= A and
A2: X in Y;
let i be object;
assume
A3: i in I;
then (union Y).i c= A.i by A1;
then
A4: union (Y.i) c= A.i by A3,Def2;
X.i in Y.i by A2,A3;
hence thesis by A4,SETFAM_1:41;
end;
theorem :: ZFMISC_1:94
for Z be ManySortedSet of I for A be non-empty ManySortedSet of I
holds (for X be ManySortedSet of I st X in A holds X c= Z) implies union A c= Z
proof
let Z be ManySortedSet of I, A be non-empty ManySortedSet of I;
assume
A1: for X be ManySortedSet of I st X in A holds X c= Z;
let i be object such that
A2: i in I;
for X9 be set st X9 in A.i holds X9 c= Z.i
proof
consider M be ManySortedSet of I such that
A3: M in A by PBOOLE:134;
let X9 be set such that
A4: X9 in A.i;
dom (M +* (i .--> X9)) = I by A2,Lm1;
then reconsider K = M +* (i .--> X9) as ManySortedSet of I by
PARTFUN1:def 2,RELAT_1:def 18;
A5: dom (i .--> X9) = {i} by FUNCOP_1:13;
i in {i} by TARSKI:def 1;
then
A6: K.i = (i .--> X9).i by A5,FUNCT_4:13
.= X9 by FUNCOP_1:72;
K in A
proof
let j be object such that
A7: j in I;
now
per cases;
case
j = i;
hence thesis by A4,A6;
end;
case
j <> i;
then not j in dom (i .--> X9) by A5,TARSKI:def 1;
then K.j = M.j by FUNCT_4:11;
hence thesis by A3,A7;
end;
end;
hence thesis;
end;
then K c= Z by A1;
hence thesis by A2,A6;
end;
then union (A.i) c= Z.i by ZFMISC_1:76;
hence thesis by A2,Def2;
end;
theorem :: ZFMISC_1:98
for B be ManySortedSet of I for A be non-empty ManySortedSet of I
st for X be ManySortedSet of I st X in A holds X (/\) B = EmptyMS I
holds union(A) (/\) B = EmptyMS I
proof
let B be ManySortedSet of I, A be non-empty ManySortedSet of I;
assume
A1: for X be ManySortedSet of I st X in A holds X (/\) B = EmptyMS I;
now
let i be object such that
A2: i in I;
for X9 be set st X9 in A.i holds X9 misses (B.i)
proof
consider M be ManySortedSet of I such that
A3: M in A by PBOOLE:134;
let X9 be set such that
A4: X9 in A.i;
dom (M +* (i .--> X9)) = I by A2,Lm1;
then reconsider K = M +* (i .--> X9) as ManySortedSet of I by
PARTFUN1:def 2,RELAT_1:def 18;
A5: dom (i .--> X9) = {i} by FUNCOP_1:13;
i in {i} by TARSKI:def 1;
then
A6: K.i = (i .--> X9).i by A5,FUNCT_4:13
.= X9 by FUNCOP_1:72;
K in A
proof
let j be object such that
A7: j in I;
now
per cases;
case
j = i;
hence thesis by A4,A6;
end;
case
j <> i;
then not j in dom (i .--> X9) by A5,TARSKI:def 1;
then K.j = M.j by FUNCT_4:11;
hence thesis by A3,A7;
end;
end;
hence thesis;
end;
then K (/\) B = EmptyMS I by A1;
then K.i /\ B.i = EmptyMS I.i by A2,PBOOLE:def 5;
then X9 /\ B.i = {} by A6,PBOOLE:5;
hence thesis by XBOOLE_0:def 7;
end;
then union (A.i) misses (B.i) by ZFMISC_1:80;
then union (A.i) /\ (B.i) = {} by XBOOLE_0:def 7;
then (union A).i /\ B.i = {} by A2,Def2;
then (union A (/\) B).i = {} by A2,PBOOLE:def 5;
hence (union(A) (/\) B).i = EmptyMS I.i by PBOOLE:5;
end;
hence thesis;
end;
theorem :: ZFMISC_1:101
for A, B be ManySortedSet of I st A (\/) B is non-empty &
for X, Y be ManySortedSet of I st X <> Y & X in A (\/) B & Y in A (\/) B
holds X (/\) Y = EmptyMS I
holds union(A (/\) B) = union A (/\) union B
proof
let A, B be ManySortedSet of I such that
A1: A (\/) B is non-empty;
assume
A2: for X, Y be ManySortedSet of I st X <> Y & X in A (\/) B & Y in A (\/) B
holds X (/\) Y = EmptyMS I;
now
let i be object such that
A3: i in I;
for X9, Y9 be set st X9 <> Y9 & X9 in A.i \/ B.i & Y9 in A.i \/ B.i
holds X9 misses Y9
proof
consider M be ManySortedSet of I such that
A4: M in A (\/) B by A1,PBOOLE:134;
A5: i in {i} by TARSKI:def 1;
let X9, Y9 be set such that
A6: X9 <> Y9 and
A7: X9 in A.i \/ B.i and
A8: Y9 in A.i \/ B.i;
dom (M +* (i .--> X9)) = I & dom (M +* (i .--> Y9)) = I by A3,Lm1;
then reconsider
Kx = M +* (i.-->X9), Ky = M +* (i.-->Y9) as ManySortedSet of
I by PARTFUN1:def 2,RELAT_1:def 18;
A9: dom (i .--> Y9) = {i} by FUNCOP_1:13;
then
A10: Ky.i = (i .--> Y9).i by A5,FUNCT_4:13
.= Y9 by FUNCOP_1:72;
A11: Ky in A (\/) B
proof
let j be object such that
A12: j in I;
now
per cases;
suppose
j = i;
hence thesis by A8,A10,A12,PBOOLE:def 4;
end;
suppose
j <> i;
then not j in dom (i .--> Y9) by A9,TARSKI:def 1;
then Ky.j = M.j by FUNCT_4:11;
hence thesis by A4,A12;
end;
end;
hence thesis;
end;
A13: dom (i .--> X9) = {i} by FUNCOP_1:13;
then
A14: Kx.i = (i .--> X9).i by A5,FUNCT_4:13
.= X9 by FUNCOP_1:72;
Kx in A (\/) B
proof
let j be object such that
A15: j in I;
now
per cases;
case
j = i;
hence thesis by A7,A14,A15,PBOOLE:def 4;
end;
case
j <> i;
then not j in dom (i .--> X9) by A13,TARSKI:def 1;
then Kx.j = M.j by FUNCT_4:11;
hence thesis by A4,A15;
end;
end;
hence thesis;
end;
then Kx (/\) Ky = EmptyMS I by A2,A6,A14,A10,A11;
then (Kx (/\) Ky).i = {} by PBOOLE:5;
then X9 /\ Y9 = {} by A3,A14,A10,PBOOLE:def 5;
hence thesis by XBOOLE_0:def 7;
end;
then union(A.i /\ B.i) = union(A.i) /\ union(B.i) by ZFMISC_1:83;
then union(A.i /\ B.i) = (union A).i /\ union(B.i) by A3,Def2;
then union(A.i /\ B.i) = (union A).i /\ (union B).i by A3,Def2;
then union(A.i /\ B.i) = (union A (/\) union B).i by A3,PBOOLE:def 5;
hence (union(A (/\) B).i) = (union A (/\) union B).i by A3,Lm7;
end;
hence thesis;
end;
theorem :: LOPCLSET:31
for A, X be ManySortedSet of I for B be non-empty ManySortedSet of I
holds (X c= union (A (\/) B) & for Y be ManySortedSet of I st Y in B
holds Y (/\) X = EmptyMS I) implies X c= union A
proof
let A, X be ManySortedSet of I, B be non-empty ManySortedSet of I;
assume that
A1: X c= union (A (\/) B) and
A2: for Y be ManySortedSet of I st Y in B holds Y (/\) X = EmptyMS I;
let i be object;
assume
A3: i in I;
A4: for Y9 be set st Y9 in B.i holds Y9 misses X.i
proof
consider M be ManySortedSet of I such that
A5: M in B by PBOOLE:134;
let Y9 be set such that
A6: Y9 in B.i;
dom (M +* (i .--> Y9)) = I by A3,Lm1;
then reconsider K = M +* (i .--> Y9) as ManySortedSet of I by
PARTFUN1:def 2,RELAT_1:def 18;
A7: dom (i .--> Y9) = {i} by FUNCOP_1:13;
i in {i} by TARSKI:def 1;
then
A8: K.i = (i .--> Y9).i by A7,FUNCT_4:13
.= Y9 by FUNCOP_1:72;
K in B
proof
let j be object such that
A9: j in I;
now
per cases;
case
j = i;
hence thesis by A6,A8;
end;
case
j <> i;
then not j in dom (i .--> Y9) by A7,TARSKI:def 1;
then K.j = M.j by FUNCT_4:11;
hence thesis by A5,A9;
end;
end;
hence thesis;
end;
then K (/\) X = EmptyMS I by A2;
then (K (/\) X).i = {} by PBOOLE:5;
then Y9 /\ X.i = {} by A3,A8,PBOOLE:def 5;
hence thesis by XBOOLE_0:def 7;
end;
X.i c= (union (A (\/) B)).i by A1,A3;
then X.i c= union (A.i \/ B.i) by A3,Lm6;
then X.i c= union (A.i) by A4,SETFAM_1:42;
hence thesis by A3,Def2;
end;
theorem :: RLVECT_3:34
for A be finite-yielding non-empty ManySortedSet of I st (for X, Y be
ManySortedSet of I st X in A & Y in A holds X c= Y or Y c= X) holds union A in
A
proof
let A be finite-yielding non-empty ManySortedSet of I such that
A1: for X, Y be ManySortedSet of I st X in A & Y in A holds X c= Y or Y c= X;
let i be object;
assume
A2: i in I;
then i in dom A by PARTFUN1:def 2;
then A.i in rng A by FUNCT_1:3;
then
A3: A.i is finite by FINSET_1:def 2;
A4: for X9, Y9 be set st X9 in A.i & Y9 in A.i holds X9 c= Y9 or Y9 c= X9
proof
let X9, Y9 be set such that
A5: X9 in A.i and
A6: Y9 in A.i;
consider M be ManySortedSet of I such that
A7: M in A by PBOOLE:134;
dom (M +* (i .--> Y9)) = I & dom (M +* (i .--> X9)) = I by A2,Lm1;
then reconsider
K1 = M +* (i.-->X9), K2 = M +* (i.-->Y9) as ManySortedSet of I
by PARTFUN1:def 2,RELAT_1:def 18;
assume
A8: not X9 c= Y9;
A9: i in {i} by TARSKI:def 1;
A10: dom (i .--> Y9) = {i} by FUNCOP_1:13;
then
A11: K2.i = (i .--> Y9).i by A9,FUNCT_4:13
.= Y9 by FUNCOP_1:72;
A12: K2 in A
proof
let j be object such that
A13: j in I;
per cases;
suppose
j = i;
hence thesis by A6,A11;
end;
suppose
j <> i;
then not j in dom (i .--> Y9) by A10,TARSKI:def 1;
then K2.j = M.j by FUNCT_4:11;
hence thesis by A7,A13;
end;
end;
A14: dom (i .--> X9) = {i} by FUNCOP_1:13;
then
A15: K1.i = (i .--> X9).i by A9,FUNCT_4:13
.= X9 by FUNCOP_1:72;
K1 in A
proof
let j be object such that
A16: j in I;
per cases;
suppose
j = i;
hence thesis by A5,A15;
end;
suppose
j <> i;
then not j in dom (i .--> X9) by A14,TARSKI:def 1;
then K1.j = M.j by FUNCT_4:11;
hence thesis by A7,A16;
end;
end;
then K1 c= K2 or K2 c= K1 by A1,A12;
hence thesis by A2,A8,A15,A11;
end;
A.i <> {} by A2,PBOOLE:def 13;
then union (A.i) in A.i by A3,A4,CARD_2:62;
hence thesis by A2,Def2;
end;