B.8 FUNCT_1

Functions and Their Basic Properties by Czeslaw Bylinski

reserve X,X1,X2,Y,Y1,Y2 for set,
p,x,x1,x2,y,y1,y2,z,z1,z2 for set;

def 1 attr X is Function-like means
for x,y1,y2 st [x,y1] in X & [x,y2] in X holds y1 = y2;

cluster Relation-like Function-like set;

mode Function is Function-like Relation-like set;

cluster empty -> Function-like set;

reserve f,f1,f2,g,g1,g2,h for Function;

2 for F being set st
(for p st p in F ex x,y st [x,y] = p) &
(for x,y1,y2 st [x,y1] in F & [x,y2] in F holds y1 = y2)
holds F is Function;

scheme GraphFunc{A()->set,P[set,set]}:
ex f st for x,y holds [x,y] in f iff x in A() & P[x,y]
provided
for x,y1,y2 st P[x,y1] & P[x,y2] holds y1 = y2;

def 4 func f.x -> set means
[x,it] in f if x in dom f otherwise it = {};

8 [x,y] in f iff x in dom f & y = f.x;
9 dom f = dom g & (for x st x in dom f holds f.x = g.x) implies f = g;

def 5 redefine func rng f means
for y holds y in it iff ex x st x in dom f & y = f.x;

12 x in dom f implies f.x in rng f;
14 dom f = {x} implies rng f = {f.x};

scheme FuncEx{A()->set,P[set,set]}:
ex f st dom f = A() & for x st x in A() holds P[x,f.x]
provided
for x,y1,y2 st x in A() & P[x,y1] & P[x,y2] holds y1 = y2 and
for x st x in A() ex y st P[x,y];

scheme Lambda{A()->set,F(set)->set}:
ex f being Function st dom f = A() & for x st x in A()
holds f.x =  F(x);

15 X <> {} implies for y ex f st dom f = X & rng f = {y};
16 (for f,g st dom f = X & dom g = X holds f = g) implies X = {};
17 dom f = dom g & rng f = {y} & rng g = {y} implies f = g;
18 Y <> {} or X = {} implies ex f st X = dom f & rng f c= Y;
19 (for y st y in Y ex x st x in dom f & y = f.x) implies Y c= rng f;

redefine func f*g;
synonym g*f;

cluster g*f -> Function-like;

20 for h st
(for x holds x in dom h iff x in dom f & f.x in dom g) &
(for x st x in dom h holds h.x = g.(f.x))
holds h = g*f;

21 x in dom(g*f) iff x in dom f & f.x in dom g;
22 x in dom(g*f) implies (g*f).x = g.(f.x);
23 x in dom f implies (g*f).x = g.(f.x);

25 z in rng(g*f) implies z in rng g;

27 dom(g*f) = dom f implies rng f c= dom g;
33 rng f c= Y & (for g,h st dom g = Y & dom h = Y & g*f = h*f
holds g = h) implies Y = rng f;

redefine func diagonal X;
synonym id X;

cluster id X -> Function-like;

34 f = id X iff dom f = X & for x st x in X holds f.x = x;
35 x in X implies (id X).x = x;
37 dom(f*(id X)) = dom f /\ X;
38 x in dom f /\ X implies f.x = (f*(id X)).x;
40 x in dom((id Y)*f) iff x in dom f & f.x in Y;
42 f*(id dom f) = f & (id rng f)*f = f;
43 (id X)*(id Y) = id(X /\ Y);
44 rng f = dom g & g*f = f implies g = id dom g;

def 8 attr f is one-to-one means
for x1,x2 st x1 in dom f & x2 in dom f & f.x1 = f.x2 holds x1 = x2;

46 f is one-to-one & g is one-to-one implies g*f is one-to-one;
47 g*f is one-to-one & rng f c= dom g implies f is one-to-one;
48 g*f is one-to-one & rng f = dom g implies f is one-to-one &
g is one-to-one;
49 f is one-to-one iff
(for g,h st rng g c= dom f & rng h c= dom f & dom g = dom h &
f*g = f*h holds g = h);
50 dom f = X & dom g = X & rng g c= X & f is one-to-one & f*g = f
implies g = id X;
51 rng(g*f) = rng g & g is one-to-one implies dom g c= rng f;
52 id X is one-to-one;
53 (ex g st g*f = id dom f) implies f is one-to-one;

cluster empty Function;

cluster empty -> one-to-one Function;

cluster one-to-one Function;

cluster f~ -> Function-like;

def 9 func f" -> Function equals f~;

54 f is one-to-one implies for g being Function holds g=f" iff
dom g = rng f &for y,x holds y in rng f & x = g.y
iff x in dom f & y = f.x;
55 f is one-to-one implies rng f = dom(f") & dom f = rng(f");
56 f is one-to-one & x in dom f implies x = (f").(f.x) & x = (f"*f).x;
57 f is one-to-one & y in rng f implies y = f.((f").y) & y = (f*f").y;
58 f is one-to-one implies dom(f"*f) = dom f & rng(f"*f) = dom f;
59 f is one-to-one implies dom(f*f") = rng f & rng(f*f") = rng f;
60 f is one-to-one & dom f = rng g & rng f = dom g &
(for x,y st x in dom f & y in dom g holds f.x = y iff g.y = x)
implies g = f";
61 f is one-to-one implies f"*f = id dom f & f*f" = id rng f;
62 f is one-to-one implies f" is one-to-one;
63 f is one-to-one & rng f = dom g & g*f = id dom f implies g = f";
64 f is one-to-one & rng g = dom f & f*g = id rng f implies g = f";
65 f is one-to-one implies (f")" = f;
66 f is one-to-one & g is one-to-one implies (g*f)" = f"*g";
67 (id X)" = (id X);

cluster f|X -> Function-like;

68 g = f|X iff dom g = dom f /\ X & for x st x in dom g
holds g.x = f.x;

70 x in dom(f|X) implies (f|X).x = f.x;
71 x in dom f /\ X implies (f|X).x = f.x;
72 x in X implies (f|X).x = f.x;
73 x in dom f & x in X implies f.x in rng(f|X);
74 X c= dom f implies dom(f|X) = X;
76 dom(f|X) c= dom f & rng(f|X) c= rng f;
82 X c= Y implies (f|X)|Y = f|X & (f|Y)|X = f|X;
84 f is one-to-one implies f|X is one-to-one;

cluster Y|f -> Function-like;

85 g = Y|f iff (for x holds x in dom g iff x in dom f & f.x in Y) &
(for x st x in dom g holds g.x = f.x);
86 x in dom(Y|f) iff x in dom f & f.x in Y;
87 x in dom(Y|f) implies (Y|f).x = f.x;
89 dom(Y|f) c= dom f & rng(Y|f) c= rng f;
97 X c= Y implies Y|(X|f) = X|f & X|(Y|f) = X|f;
99 f is one-to-one implies Y|f is one-to-one;

def 12 func f.:X means
for y holds y in it iff ex x st x in dom f & x in X & y = f.x;

117 x in dom f implies f.:{x} = {f.x};
118 x1 in dom f & x2 in dom f implies f.:{x1,x2} = {f.x1,f.x2};
120 (Y|f).:X c= f.:X;
121 f is one-to-one implies f.:(X1 /\ X2) = f.:X1 /\ f.:X2;
122 (for X1,X2 holds f.:(X1 /\ X2) = f.:X1 /\ f.:X2)
implies f is one-to-one;
123 f is one-to-one implies f.:(X1 \ X2) = f.:X1 \ f.:X2;
124 (for X1,X2 holds f.:(X1 \ X2) = f.:X1 \ f.:X2)
implies f is one-to-one;
125 X misses Y & f is one-to-one implies f.:X misses f.:Y;
126 (Y|f).:X = Y /\ f.:X;

def 13 redefine func f"Y means
for x holds x in it iff x in dom f & f.x in Y;

137 f"(Y1 /\ Y2) = f"Y1 /\ f"Y2;
138 f"(Y1 \ Y2) = f"Y1 \ f"Y2;
139 (f|X)"Y = X /\ (f"Y);
142 y in rng f iff f"{y} <> {};
143 (for y st y in Y holds f"{y} <> {}) implies Y c= rng f;
144 (for y st y in rng f ex x st f"{y} = {x}) iff f is one-to-one;
145 f.:(f"Y) c= Y;
146 X c= dom f implies X c= f"(f.:X);
147 Y c= rng f implies f.:(f"Y) = Y;
148 f.:(f"Y) = Y /\ f.:(dom f);
149 f.:(X /\ f"Y) c= (f.:X) /\ Y;
150 f.:(X /\ f"Y) = (f.:X) /\ Y;
151 X /\ f"Y c= f"(f.:X /\ Y);
152 f is one-to-one implies f"(f.:X) c= X;
153 (for X holds f"(f.:X) c= X) implies f is one-to-one;
154 f is one-to-one implies f.:X = (f")"X;
155 f is one-to-one implies f"Y = (f").:Y;
156 Y = rng f & dom g = Y & dom h = Y & g*f = h*f implies g = h;
157 f.:X1 c= f.:X2 & X1 c= dom f & f is one-to-one implies X1 c= X2;
158 f"Y1 c= f"Y2 & Y1 c= rng f implies Y1 c= Y2;
159 f is one-to-one iff for y ex x st f"{y} c= {x};
160 rng f c= dom g implies f"X c= (g*f)"(g.:X);