B.11 FUNCT_2
Functions from a Set to a Set by Czeslaw Bylinski
reserve P,Q,X,Y,Y1,Y2,Z,p,x,x',x1,x2,y,y1,y2,z for set;
def 1 attr R is quasi_total means X = dom R if Y = {} implies X = {}
otherwise R = {};
cluster quasi_total Function-like Relation of X,Y;
cluster total -> quasi_total PartFunc of X,Y;
mode Function of X,Y is quasi_total Function-like Relation of X,Y;
3 for f being Function holds f is Function of dom f, rng f;
4 for f being Function st rng f c= Y holds f is Function of dom f, Y;
5 for f being Function st dom f = X & for x st x in X
holds f.x in Y holds f is Function of X,Y;
6 for f being Function of X,Y st Y <> {} & x in X holds f.x in rng f;
7 for f being Function of X,Y st Y <> {} & x in X holds f.x in Y;
8 for f being Function of X,Y st (Y = {} implies X = {}) & rng f c=
Z
holds f is Function of X,Z;
9 for f being Function of X,Y
st (Y = {} implies X = {}) & Y c= Z holds f is Function of X,Z;
scheme FuncEx1{X, Y() -> set, P[set,set]}:
ex f being Function of X(),Y() st for x st x in X() holds P[x,f.x]
provided
for x st x in X() ex y st y in Y() & P[x,y];
scheme Lambda1{X, Y() -> set, F(set)->set}:
ex f being Function of X(),Y() st for x st x in X() holds f.x = F(x)
provided
for x st x in X() holds F(x) in Y();
def 2 func Funcs(X,Y) -> set means
x in it iff ex f being Function st x = f & dom
f = X & rng f c= Y;
11 for f being Function of X,Y st Y = {} implies X = {}
holds f in Funcs(X,Y);
12 for f being Function of X,X holds f in Funcs(X,X);
14 X <> {} implies Funcs(X,{}) = {};
16 for f being Function of X,Y
st Y <> {} & for y st y in Y ex x st x in
X & y = f.x
holds rng f = Y;
17 for f being Function of X,Y st y in Y &
rng f = Y ex x st x in X & f.x = y;
18 for f1,f2 being Function of X,Y
st for x st x in X holds f1.x = f2.x
holds f1 = f2;
19 for f being Function of X,Y for g being Function of Y,Z
st Y = {} implies Z = {} or X = {}
holds g*f is Function of X,Z;
20 for f being Function of X,Y for g being Function of Y,Z
st Y <> {} & Z <> {} & rng f =
Y & rng g = Z holds rng(g*f) = Z;
21 for f being Function of X,Y, g being Function
st Y <> {} & x in X holds (g*f).x = g.(f.x);
22 for f being Function of X,Y st Y <> {} holds rng f = Y iff
for Z st Z <> {} for g,h being Function of Y,Z
st g*f = h*f
holds g = h;
23 for f being Function of X,Y
st Y = {} implies X = {} holds f*(id X) = f & (id
Y)*f = f;
24 for f being Function of X,Y for g being Function of Y,X
st f*g = id Y holds rng f = Y;
25 for f being Function of X,Y st Y = {} implies X = {}
holds f is one-to-one iff
for x1,x2 st x1 in X & x2 in X & f.x1 = f.x2
holds x1 = x2;
for f being Function of X,Y for g being Function of
Y,Z
st (Z = {} implies Y = {}) & (Y = {}
implies X = {}) & g*f is one-to-one
holds f is one-to-one;
27 for f being Function of X,Y st X <> {} & Y <> {}
holds f is one-to-one iff
for Z for g,h being Function of Z,X st f*g = f*h holds
g = h;
28 for f being Function of X,Y for g being Function of Y,Z
st Z <> {} & rng(g*f) = Z & g is one-to-one
holds rng f = Y;
29 for f being Function of X,Y for g being Function of Y,X
st Y <> {} & g*f = id X holds f is one-to-one
& rng g = X;
30 for f being Function of X,Y for g being Function of Y,Z
st (Z = {} implies Y = {}) & g*f is one-to-one
& rng f = Y
holds f is one-to-one & g is one-to-one;
31 for f being Function of X,Y st f is one-to-one & rng f = Y
holds f" is Function of Y,X;
32 for f being Function of X,Y
st Y <> {} & f is one-to-one & x in X
holds (f").(f.x) = x;
34 for f being Function of X,Y for g being Function of Y,X
st X <> {} & Y <> {} & rng f =
Y & f is one-to-one &
for y,x holds y in Y & g.y = x iff x in X &
f.x = y
holds g = f";
35 for f being Function of X,Y
st Y <> {} & rng f = Y & f is one-to-one
holds f"*f = id X & f*f" = id Y;
36 for f being Function of X,Y for g being Function of Y,X
st X <> {} & Y <> {} & rng f =
Y & g*f = id X & f is one-to-one
holds g = f";
37 for f being Function of X,Y st Y <> {}
& ex g being Function of Y,X st g*f = id X
holds f is one-to-one;
38 for f being Function of X,Y
st (Y = {} implies X = {}) & Z c= X holds f|Z
is Function of Z,Y;
40 for f being Function of X,Y st X c= Z holds f|Z = f;
41 for f being Function of X,Y st Y <> {} & x in X & f.x in
Z holds (Z|f).x = f.x;
42 for f being Function of X,Y st (Y = {} implies X = {}) & Y c= Z
holds Z|f = f;
43 for f being Function of X,Y st Y <> {}
for y holds y in f.:P iff ex x st x in X & x in
P & y = f.x;
44 for f being Function of X,Y holds f.:P c= Y;
redefine func f.:P -> Subset of Y;
45 for f being Function of X,Y st Y = {} implies X = {}
holds f.:X = rng f;
46 for f being Function of X,Y
st Y <> {} for x holds x in f"Q iff x in X & f.x in Q;
47 for f being PartFunc of X,Y holds f"Q c= X;
redefine func f"Q -> Subset of X;
48 for f being Function of X,Y st Y = {} implies X = {}
holds f"Y = X;
49 for f being Function of X,Y
holds (for y st y in Y holds f"{y} <> {})
iff rng f = Y;
50 for f being Function of X,Y
st (Y = {} implies X = {}) & P c= X holds P c=
f"(f.:P);
51 for f being Function of X,Y st Y = {} implies X = {}
holds f"(f.:X) = X;
53 for f being Function of X,Y for g being Function of Y,Z
st (Z = {} implies Y = {}) & (Y = {} implies X
= {})
holds f"Q c= (g*f)"(g.:Q);
54 for f being Function of {},Y holds dom f = {} & rng f = {};
55 for f being Function st dom f = {} holds f is Function of {},Y;
56 for f1 being Function of {},Y1 for f2 being Function of {},Y2
holds f1 = f2;
58 for f being Function of {},Y holds f is one-to-one;
59 for f being Function of {},Y holds f.:P = {};
60 for f being Function of {},Y holds f"Q = {};
61 for f being Function of {x},Y st Y <> {} holds f.x in Y;
62 for f being Function of {x},Y st Y <> {} holds rng f = {f.x};
63 for f being Function of {x},Y st Y <> {} holds f is one-to-one;
64 for f being Function of {x},Y st Y <> {} holds f.:P c= {f.x};
65 for f being Function of X,{y} st x in X holds f.x = y;
66 for f1,f2 being Function of X,{y} holds f1 = f2;
redefine func g*f -> Function of X,X;
redefine func id X -> Function of X,X;
67 for f being Function of X,X holds dom f = X & rng f c= X;
70 for f being Function of X,X, g being Function
st x in X holds (g*f).x = g.(f.x);
73 for f,g being Function of X,X st rng f = X & rng g = X
holds rng(g*f) = X;
74 for f being Function of X,X holds f*(id X) = f & (id X)*f = f;
75 for f,g being Function of X,X st g*f = f & rng f = X
holds g = id X;
76 for f,g being Function of X,X st f*g = f & f is one-to-one
holds g = id X;
77 for f being Function of X,X holds f is one-to-one iff
for x1,x2 st x1 in X & x2 in X & f.x1 = f.x2
holds x1 = x2;
79 for f being Function of X,X holds f.:X = rng f;
82 for f being Function of X,X holds f"(f.:X) = X;
def 3 attr f is onto means rng f = Y;
def 4 attr f is bijective means f is one-to-one onto;
cluster bijective -> one-to-one onto Function of X,Y;
cluster one-to-one onto -> bijective Function of X,Y;
cluster bijective Function of X,X;
mode Permutation of X is bijective Function of X,X;
83 for f being Function of X, X holds
f is Permutation of X iff f is one-to-one & rng
f = X;
85 for f being Function of X,X st f is one-to-one holds
for x1,x2 st x1 in X & x2 in X & f.x1 = f.x2
holds x1 = x2;
redefine func g*f -> Permutation of X;
redefine func id X -> Permutation of X;
redefine func f" -> Permutation of X;
86 for f,g being Permutation of X st g*f = g holds f = id X;
87 for f,g being Permutation of X st g*f = id X holds g = f";
88 for f being Permutation of X holds (f")*f =id X & f*(f") = id X;
92 for f being Permutation of X st P c= X
holds f.:(f"P) = P & f"(f.:P) = P;
93 for f being Function of X,X st f is one-to-one
holds f.:P = (f")"P & f"P = (f").:P;
reserve C,D,E for non empty set;
cluster quasi_total -> total PartFunc of X,D;
redefine func g*f -> Function of X,Z;
reserve c for Element of C;
reserve d for Element of D;
redefine func f.c -> Element of D;
scheme FuncExD{C, D() -> non empty set, P[set,set]}:
ex f being Function of C(),D() st for x being Element of C()
holds P[x,f.x]
provided
for x being Element of C() ex y being Element of D() st P[x,y];
scheme LambdaD{C, D() -> non empty set,
F(Element of C()) -> Element of D()}:
ex f being Function of C(),D() st
for x being Element of C() holds f.x = F(x);
113 for f1,f2 being Function of X,Y st
for x being Element of X holds f1.x = f2.x holds f1 = f2;
116 for f being Function of C,D for d
holds d in f.:P iff ex c st c in P & d =
f.c;
118 for f1,f2 being Function of [:X,Y:],Z
st for x,y st x in X & y in Y holds f1.[x,y]
= f2.[x,y]
holds f1 = f2;
119 for f being Function of [:X,Y:],Z st x in X & y in Y & Z <> {}
holds f.[x,y] in Z;
scheme FuncEx2{X, Y, Z() -> set, P[set,set,set]}:
ex f being Function of [:X(),Y():],Z() st
for x,y st x in X() & y in Y() holds P[x,y,f.[x,y]]
provided
for x,y st x in X() & y in Y() ex z st z in Z() & P[x,y,z];
scheme Lambda2{X, Y, Z() -> set, F(set,set)->set}:
ex f being Function of [:X(),Y():],Z()
st for x,y st x in X() & y in Y() holds f.[x,y] = F(x,y)
provided
for x,y st x in X() & y in Y() holds F(x,y) in Z();
120 for f1,f2 being Function of [:C,D:],E st for c,d
holds f1.[c,d] = f2.[c,d] holds f1 = f2;
scheme FuncEx2D{X, Y, Z() -> non empty set, P[set,set,set]}:
ex f being Function of [:X(),Y():],Z() st
for x being Element of X() for y being Element of Y() holds P[x,y,f.[x,y]]
provided
for x being Element of X() for y being Element of Y()
ex z being Element of Z() st P[x,y,z];
scheme Lambda2D{X, Y, Z() -> non empty set,
F(Element of X(),Element of Y()) -> Element of Z()}:
ex f being Function of [:X(),Y():],Z()
st for x being Element of X() for y being Element of Y()
holds f.[x,y]=F(x,y);
121 for f being set st f in Funcs(X,Y) holds f is Function of X,Y;
scheme Lambda1C{A, B() -> set, C[set], F(set)->set, G(set)->set}:
ex f being Function of A(),B() st
for x st x in A() holds
(C[x] implies f.x = F(x)) & (not C[x] implies f.x = G(x))
provided
for x st x in A() holds
(C[x] implies F(x) in B()) & (not C[x] implies G(x) in B());
123 for f being Function of {},Y holds f = {};
124 for f being Function of X,Y st f is one-to-one
holds f" is PartFunc of Y,X;
125 for f being Function of X,X st f is one-to-one
holds f" is PartFunc of X,X;
127 for f being Function of X,Y st Y = {}
implies X = {} holds <:f,X,Y:> = f;
128 for f being Function of X,X holds <:f,X,X:> = f;
130 for f being PartFunc of X,Y st dom f = X
holds f is Function of X,Y;
131 for f being PartFunc of X,Y st f is total
holds f is Function of X,Y;
132 for f being PartFunc of X,Y st (Y = {}
implies X = {}) & f is Function of X,Y holds
f is total;
133 for f being Function of X,Y
st (Y = {} implies X = {}) holds <:f,X,Y:> is total;
134 for f being Function of X,X holds <:f,X,X:> is total;
136 for f being PartFunc of X,Y st Y = {} implies X = {}
ex g being Function of X,Y st for x st x in dom f
holds g.x = f.x;
141 Funcs(X,Y) c= PFuncs(X,Y);
142 for f,g being Function of X,Y st (Y = {}
implies X = {}) & f tolerates g holds f =
g;
143 for f,g being Function of X,X st f tolerates g holds f = g;
145 for f being PartFunc of X,Y for g being Function of X,Y
st Y = {} implies X = {}
holds f tolerates g iff for x st x in dom f holds
f.x = g.x;
146 for f being PartFunc of X,X for g being Function of X,X
holds f tolerates g iff for x st x in dom f holds f.x = g.x;
148 for f being PartFunc of X,Y st Y = {} implies X = {}
ex g being Function of X,Y st f tolerates g;
149 for f being PartFunc of X,X ex g being Function of X,X st
f tolerates g;
151 for f,g being PartFunc of X,Y for h being Function of X,Y
st (Y = {} implies X = {}) & f tolerates h & g tolerates h
holds f tolerates g;
152 for f,g being PartFunc of X,X for h being Function of X,X
st f tolerates h & g tolerates h holds f
tolerates g;
154 for f,g being PartFunc of X,Y st (Y = {}
implies X = {}) & f tolerates g
ex h being Function of X,Y st f tolerates h &
g tolerates h;
155 for f being PartFunc of X,Y for g being Function of X,Y
st (Y = {} implies X = {}) & f tolerates
g
holds g in TotFuncs f;
156 for f being PartFunc of X,X for g being Function of X,X
st f tolerates g holds g in TotFuncs f;
158 for f being PartFunc of X,Y for g being set
st g in TotFuncs(f) holds g is Function of X,Y;
159 for f being PartFunc of X,Y holds TotFuncs f c= Funcs(X,Y);
160 TotFuncs <:{},X,Y:> = Funcs(X,Y);
161 for f being Function of X,Y st Y = {} implies X = {}
holds TotFuncs <:f,X,Y:> = {f};
162 for f being Function of X,X holds TotFuncs <:f,X,X:> = {f};
164 for f being PartFunc of X,{y} for g being Function of X,{y}
holds TotFuncs f = {g};
165 for f,g being PartFunc of X,Y
st g c= f holds TotFuncs f c= TotFuncs g;
166 for f,g being PartFunc of X,Y
st dom g c= dom f & TotFuncs f c= TotFuncs g holds g c= f;
167 for f,g being PartFunc of X,Y
st TotFuncs f c= TotFuncs g & (for y holds
Y <> {y})
holds g c= f;
168 for f,g being PartFunc of X,Y
st (for y holds Y <> {y}) & TotFuncs f =
TotFuncs g holds f = g;