FUNCT_6: Cartesian Product of Functions

:: Cartesian Product of Functions
:: by Grzegorz Bancerek
::
:: Received September 30, 1991
:: Copyright (c) 1991 Association of Mizar Users

begin

theorem Th1: :: FUNCT_6:1

for x, X being set holds
( x in product <*X*> iff ex y being set st
( y in X & x = <*y*> ) )

proof end;

theorem Th2: :: FUNCT_6:2

for z, X, Y being set holds
( z in product <*X,Y*> iff ex x, y being set st
( x in X & y in Y & z = <*x,y*> ) )

proof end;

theorem Th3: :: FUNCT_6:3

for a, X, Y, Z being set holds
( a in product <*X,Y,Z*> iff ex x, y, z being set st
( x in X & y in Y & z in Z & a = <*x,y,z*> ) )

proof end;

theorem :: FUNCT_6:4

for D being non empty set holds product <*D*> = 1 -tuples_on D

proof end;

theorem Th5: :: FUNCT_6:5

for D1, D2 being non empty set holds product <*D1,D2*> = { <*d1,d2*> where d1 is Element of D1, d2 is Element of D2 : verum }

proof end;

theorem :: FUNCT_6:6

for D being non empty set holds product <*D,D*> = 2 -tuples_on D

proof end;

theorem Th7: :: FUNCT_6:7

for D1, D2, D3 being non empty set holds product <*D1,D2,D3*> = { <*d1,d2,d3*> where d1 is Element of D1, d2 is Element of D2, d3 is Element of D3 : verum }

proof end;

theorem :: FUNCT_6:8

for D being non empty set holds product <*D,D,D*> = 3 -tuples_on D

proof end;

theorem :: FUNCT_6:9

for i being Element of NAT
for D being non empty set holds product (i |-> D) = i -tuples_on D

proof end;

theorem Th10: :: FUNCT_6:10

for f being Function holds product f c= Funcs ((dom f),(Union f))

proof end;

begin

theorem Th11: :: FUNCT_6:11

for x being set
for f being Function st x in dom (~ f) holds
ex y, z being set st x = [y,z]

proof end;

theorem Th12: :: FUNCT_6:12

for X, Y, z being set holds ~ ([:X,Y:] --> z) = [:Y,X:] --> z

proof end;

theorem Th13: :: FUNCT_6:13

for f being Function holds
( curry f = curry' (~ f) & uncurry f = ~ (uncurry' f) )

proof end;

theorem :: FUNCT_6:14

for X, Y, z being set st [:X,Y:] <> {} holds
( curry ([:X,Y:] --> z) = X --> (Y --> z) & curry' ([:X,Y:] --> z) = Y --> (X --> z) )

proof end;

theorem :: FUNCT_6:15

for X, Y, z being set holds
( uncurry (X --> (Y --> z)) = [:X,Y:] --> z & uncurry' (X --> (Y --> z)) = [:Y,X:] --> z )

proof end;

theorem Th16: :: FUNCT_6:16

for x being set
for f, g being Function st x in dom f & g = f . x holds
( rng g c= rng (uncurry f) & rng g c= rng (uncurry' f) )

proof end;

theorem Th17: :: FUNCT_6:17

for X being set
for f being Function holds
( dom (uncurry (X --> f)) = [:X,(dom f):] & rng (uncurry (X --> f)) c= rng f & dom (uncurry' (X --> f)) = [:(dom f),X:] & rng (uncurry' (X --> f)) c= rng f )

proof end;

theorem :: FUNCT_6:18

for X being set
for f being Function st X <> {} holds
( rng (uncurry (X --> f)) = rng f & rng (uncurry' (X --> f)) = rng f )

proof end;

theorem Th19: :: FUNCT_6:19

for X, Y, Z being set
for f being Function st [:X,Y:] <> {} & f in Funcs ([:X,Y:],Z) holds
( curry f in Funcs (X,(Funcs (Y,Z))) & curry' f in Funcs (Y,(Funcs (X,Z))) )

proof end;

theorem Th20: :: FUNCT_6:20

for X, Y, Z being set
for f being Function st f in Funcs (X,(Funcs (Y,Z))) holds
( uncurry f in Funcs ([:X,Y:],Z) & uncurry' f in Funcs ([:Y,X:],Z) )

proof end;

theorem :: FUNCT_6:21

for X, Y, Z, V1, V2 being set
for f being Function st ( curry f in Funcs (X,(Funcs (Y,Z))) or curry' f in Funcs (Y,(Funcs (X,Z))) ) & dom f c= [:V1,V2:] holds
f in Funcs ([:X,Y:],Z)

proof end;

theorem :: FUNCT_6:22

for X, Y, Z, V1, V2 being set
for f being Function st ( uncurry f in Funcs ([:X,Y:],Z) or uncurry' f in Funcs ([:Y,X:],Z) ) & rng f c= PFuncs (V1,V2) & dom f = X holds
f in Funcs (X,(Funcs (Y,Z)))

proof end;

theorem :: FUNCT_6:23

for X, Y, Z being set
for f being Function st f in PFuncs ([:X,Y:],Z) holds
( curry f in PFuncs (X,(PFuncs (Y,Z))) & curry' f in PFuncs (Y,(PFuncs (X,Z))) )

proof end;

theorem Th24: :: FUNCT_6:24

for X, Y, Z being set
for f being Function st f in PFuncs (X,(PFuncs (Y,Z))) holds
( uncurry f in PFuncs ([:X,Y:],Z) & uncurry' f in PFuncs ([:Y,X:],Z) )

proof end;

theorem :: FUNCT_6:25

for X, Y, Z, V1, V2 being set
for f being Function st ( curry f in PFuncs (X,(PFuncs (Y,Z))) or curry' f in PFuncs (Y,(PFuncs (X,Z))) ) & dom f c= [:V1,V2:] holds
f in PFuncs ([:X,Y:],Z)

proof end;

theorem :: FUNCT_6:26

for X, Y, Z, V1, V2 being set
for f being Function st ( uncurry f in PFuncs ([:X,Y:],Z) or uncurry' f in PFuncs ([:Y,X:],Z) ) & rng f c= PFuncs (V1,V2) & dom f c= X holds
f in PFuncs (X,(PFuncs (Y,Z)))

proof end;

begin

definition

let X be set ;
func SubFuncs X -> set means :Def1: :: FUNCT_6:def 1
for x being set holds
( x in it iff ( x in X & x is Function ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines SubFuncs FUNCT_6:def 1 :

theorem Th27: :: FUNCT_6:27

for X being set holds SubFuncs X c= X

proof end;

theorem Th28: :: FUNCT_6:28

for x being set
for f being Function holds
( x in f " (SubFuncs (rng f)) iff ( x in dom f & f . x is Function ) )

proof end;

Lm1: for X being set st ( for x being set st x in X holds
x is Function ) holds
SubFuncs X = X

proof end;

theorem Th29: :: FUNCT_6:29

for f, g, h being Function holds
( SubFuncs {} = {} & SubFuncs {f} = {f} & SubFuncs {f,g} = {f,g} & SubFuncs {f,g,h} = {f,g,h} )

proof end;

theorem :: FUNCT_6:30

for Y, X being set st Y c= SubFuncs X holds
SubFuncs Y = Y

proof end;

definition

let f be Function;
func doms f -> Function means :Def2: :: FUNCT_6:def 2
( dom it = f " (SubFuncs (rng f)) & ( for x being set st x in f " (SubFuncs (rng f)) holds
it . x = proj1 (f . x) ) );
existence

proof end;

uniqueness

proof end;

func rngs f -> Function means :Def3: :: FUNCT_6:def 3
( dom it = f " (SubFuncs (rng f)) & ( for x being set st x in f " (SubFuncs (rng f)) holds
it . x = proj2 (f . x) ) );
existence

proof end;

uniqueness

proof end;

func meet f -> set equals :: FUNCT_6:def 4
meet (rng f);
correctness ;

end;

:: deftheorem Def2 defines doms FUNCT_6:def 2 :

:: deftheorem Def3 defines rngs FUNCT_6:def 3 :

:: deftheorem defines meet FUNCT_6:def 4 :

theorem Th31: :: FUNCT_6:31

for x being set
for f, g being Function st x in dom f & g = f . x holds
( x in dom (doms f) & (doms f) . x = dom g & x in dom (rngs f) & (rngs f) . x = rng g )

proof end;

theorem :: FUNCT_6:32

( doms {} = {} & rngs {} = {} )

proof end;

theorem Th33: :: FUNCT_6:33

for f being Function holds
( doms <*f*> = <*(dom f)*> & rngs <*f*> = <*(rng f)*> )

proof end;

theorem Th34: :: FUNCT_6:34

for f, g being Function holds
( doms <*f,g*> = <*(dom f),(dom g)*> & rngs <*f,g*> = <*(rng f),(rng g)*> )

proof end;

theorem :: FUNCT_6:35

for f, g, h being Function holds
( doms <*f,g,h*> = <*(dom f),(dom g),(dom h)*> & rngs <*f,g,h*> = <*(rng f),(rng g),(rng h)*> )

proof end;

theorem Th36: :: FUNCT_6:36

for X being set
for f being Function holds
( doms (X --> f) = X --> (dom f) & rngs (X --> f) = X --> (rng f) )

proof end;

theorem Th37: :: FUNCT_6:37

for x being set
for f being Function st f <> {} holds
( x in meet f iff for y being set st y in dom f holds
x in f . y )

proof end;

theorem :: FUNCT_6:38

canceled;

theorem Th39: :: FUNCT_6:39

for X being set holds
( Union <*X*> = X & meet <*X*> = X )

proof end;

theorem Th40: :: FUNCT_6:40

for X, Y being set holds
( Union <*X,Y*> = X \/ Y & meet <*X,Y*> = X /\ Y )

proof end;

theorem :: FUNCT_6:41

for X, Y, Z being set holds
( Union <*X,Y,Z*> = (X \/ Y) \/ Z & meet <*X,Y,Z*> = (X /\ Y) /\ Z )

proof end;

theorem :: FUNCT_6:42

for Y being set holds
( Union ({} --> Y) = {} & meet ({} --> Y) = {} )

proof end;

theorem Th43: :: FUNCT_6:43

for X, Y being set st X <> {} holds
( Union (X --> Y) = Y & meet (X --> Y) = Y )

proof end;

definition

let f be Function;
let x, y be set ;
func f .. (x,y) -> set equals :: FUNCT_6:def 5
(uncurry f) . (x,y);
correctness ;

end;

:: deftheorem defines .. FUNCT_6:def 5 :

theorem :: FUNCT_6:44

canceled;

theorem :: FUNCT_6:45

for x being set
for f, g, h being Function st x in dom f holds
( <*f*> .. (1,x) = f . x & <*f,g*> .. (1,x) = f . x & <*f,g,h*> .. (1,x) = f . x )

proof end;

theorem :: FUNCT_6:46

for x being set
for g, f, h being Function st x in dom g holds
( <*f,g*> .. (2,x) = g . x & <*f,g,h*> .. (2,x) = g . x )

proof end;

theorem :: FUNCT_6:47

for x being set
for h, f, g being Function st x in dom h holds
<*f,g,h*> .. (3,x) = h . x

proof end;

theorem :: FUNCT_6:48

for x, X, y being set
for f being Function st x in X & y in dom f holds
(X --> f) .. (x,y) = f . y

proof end;

begin

definition

let f be Function;
func <:f:> -> Function equals :: FUNCT_6:def 6
curry ((uncurry' f) | [:(meet (doms f)),(dom f):]);
correctness ;

end;

:: deftheorem defines <: FUNCT_6:def 6 :

theorem Th49: :: FUNCT_6:49

for f being Function holds
( dom <:f:> = meet (doms f) & rng <:f:> c= product (rngs f) )

proof end;

theorem Th50: :: FUNCT_6:50

for x being set
for f being Function st x in dom <:f:> holds
<:f:> . x is Function

proof end;

theorem Th51: :: FUNCT_6:51

for x being set
for f, g being Function st x in dom <:f:> & g = <:f:> . x holds
( dom g = f " (SubFuncs (rng f)) & ( for y being set st y in dom g holds
( [y,x] in dom (uncurry f) & g . y = (uncurry f) . (y,x) ) ) )

proof end;

theorem Th52: :: FUNCT_6:52

for x being set
for f being Function st x in dom <:f:> holds
for g being Function st g in rng f holds
x in dom g

proof end;

theorem :: FUNCT_6:53

for x being set
for g, f being Function st g in rng f & ( for g being Function st g in rng f holds
x in dom g ) holds
x in dom <:f:>

proof end;

theorem Th54: :: FUNCT_6:54

for x, y being set
for f, g, h being Function st x in dom f & g = f . x & y in dom <:f:> & h = <:f:> . y holds
g . y = h . x

proof end;

theorem :: FUNCT_6:55

for x, y being set
for f being Function st x in dom f & f . x is Function & y in dom <:f:> holds
f .. (x,y) = <:f:> .. (y,x)

proof end;

definition

let f be Function;
func Frege f -> Function means :Def7: :: FUNCT_6:def 7
( dom it = product (doms f) & ( for g being Function st g in product (doms f) holds
ex h being Function st
( it . g = h & dom h = f " (SubFuncs (rng f)) & ( for x being set st x in dom h holds
h . x = (uncurry f) . (x,(g . x)) ) ) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines Frege FUNCT_6:def 7 :

theorem :: FUNCT_6:56

for x being set
for g, f being Function st g in product (doms f) & x in dom g holds
(Frege f) .. (g,x) = f .. (x,(g . x))

proof end;

Lm2: for f being Function holds rng (Frege f) c= product (rngs f)

proof end;

theorem Th57: :: FUNCT_6:57

for x being set
for f, g, h, h9 being Function st x in dom f & g = f . x & h in product (doms f) & h9 = (Frege f) . h holds
( h . x in dom g & h9 . x = g . (h . x) & h9 in product (rngs f) )

proof end;

Lm3: for f being Function holds product (rngs f) c= rng (Frege f)

proof end;

theorem Th58: :: FUNCT_6:58

for f being Function holds rng (Frege f) = product (rngs f)

proof end;

theorem Th59: :: FUNCT_6:59

for f being Function st not {} in rng f holds
( Frege f is one-to-one iff for g being Function st g in rng f holds
g is one-to-one )

proof end;

begin

theorem :: FUNCT_6:60

( <:{}:> = {} & Frege {} = {} .--> {} )

proof end;

theorem :: FUNCT_6:61

for h being Function holds
( dom <:<*h*>:> = dom h & ( for x being set st x in dom h holds
<:<*h*>:> . x = <*(h . x)*> ) )

proof end;

theorem Th62: :: FUNCT_6:62

for f1, f2 being Function holds
( dom <:<*f1,f2*>:> = (dom f1) /\ (dom f2) & ( for x being set st x in (dom f1) /\ (dom f2) holds
<:<*f1,f2*>:> . x = <*(f1 . x),(f2 . x)*> ) )

proof end;

theorem :: FUNCT_6:63

for X being set
for f being Function st X <> {} holds
( dom <:(X --> f):> = dom f & ( for x being set st x in dom f holds
<:(X --> f):> . x = X --> (f . x) ) )

proof end;

theorem :: FUNCT_6:64

for h being Function holds
( dom (Frege <*h*>) = product <*(dom h)*> & rng (Frege <*h*>) = product <*(rng h)*> & ( for x being set st x in dom h holds
(Frege <*h*>) . <*x*> = <*(h . x)*> ) )

proof end;

theorem Th65: :: FUNCT_6:65

for f1, f2 being Function holds
( dom (Frege <*f1,f2*>) = product <*(dom f1),(dom f2)*> & rng (Frege <*f1,f2*>) = product <*(rng f1),(rng f2)*> & ( for x, y being set st x in dom f1 & y in dom f2 holds
(Frege <*f1,f2*>) . <*x,y*> = <*(f1 . x),(f2 . y)*> ) )

proof end;

theorem :: FUNCT_6:66

for X being set
for f being Function holds
( dom (Frege (X --> f)) = Funcs (X,(dom f)) & rng (Frege (X --> f)) = Funcs (X,(rng f)) & ( for g being Function st g in Funcs (X,(dom f)) holds
(Frege (X --> f)) . g = f * g ) )

proof end;

theorem :: FUNCT_6:67

for x being set
for f1, f2 being Function st x in dom f1 & x in dom f2 holds
for y1, y2 being set holds
( <:f1,f2:> . x = [y1,y2] iff <:<*f1,f2*>:> . x = <*y1,y2*> )

proof end;

theorem :: FUNCT_6:68

for x, y being set
for f1, f2 being Function st x in dom f1 & y in dom f2 holds
for y1, y2 being set holds
( [:f1,f2:] . (x,y) = [y1,y2] iff (Frege <*f1,f2*>) . <*x,y*> = <*y1,y2*> )

proof end;

theorem Th69: :: FUNCT_6:69

for X being set
for f, g being Function st dom f = X & dom g = X & ( for x being set st x in X holds
f . x,g . x are_equipotent ) holds
product f, product g are_equipotent

proof end;

theorem Th70: :: FUNCT_6:70

for f, h, g being Function st dom f = dom h & dom g = rng h & h is one-to-one & ( for x being set st x in dom h holds
f . x,g . (h . x) are_equipotent ) holds
product f, product g are_equipotent

proof end;

theorem :: FUNCT_6:71

for X being set
for f being Function
for P being Permutation of X st dom f = X holds
product f, product (f * P) are_equipotent

proof end;

begin

definition

let f be Function;
let X be set ;
func Funcs (f,X) -> Function means :Def8: :: FUNCT_6:def 8
( dom it = dom f & ( for x being set st x in dom f holds
it . x = Funcs ((f . x),X) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines Funcs FUNCT_6:def 8 :

theorem :: FUNCT_6:72

for f being Function st not {} in rng f holds
Funcs (f,{}) = (dom f) --> {}

proof end;

theorem :: FUNCT_6:73

for X being set holds Funcs ({},X) = {}

proof end;

theorem :: FUNCT_6:74

for X, Y being set holds Funcs (<*X*>,Y) = <*(Funcs (X,Y))*>

proof end;

theorem :: FUNCT_6:75

for X, Y, Z being set holds Funcs (<*X,Y*>,Z) = <*(Funcs (X,Z)),(Funcs (Y,Z))*>

proof end;

theorem :: FUNCT_6:76

for X, Y, Z being set holds Funcs ((X --> Y),Z) = X --> (Funcs (Y,Z))

proof end;

Lm4: for x, y, z being set
for f, g being Function st [x,y] in dom f & g = (curry f) . x & z in dom g holds
[x,z] in dom f

proof end;

theorem :: FUNCT_6:77

for X being set
for f being Function holds Funcs ((Union (disjoin f)),X), product (Funcs (f,X)) are_equipotent

proof end;

definition

let X be set ;
let f be Function;
func Funcs (X,f) -> Function means :Def9: :: FUNCT_6:def 9
( dom it = dom f & ( for x being set st x in dom f holds
it . x = Funcs (X,(f . x)) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines Funcs FUNCT_6:def 9 :

Lm5: for X being set
for f being Function st f <> {} & X <> {} holds
product (Funcs (X,f)), Funcs (X,(product f)) are_equipotent

proof end;

theorem Th78: :: FUNCT_6:78

for f being Function holds Funcs ({},f) = (dom f) --> {{}}

proof end;

theorem Th79: :: FUNCT_6:79

for X being set holds Funcs (X,{}) = {}

proof end;

theorem :: FUNCT_6:80

for X, Y being set holds Funcs (X,<*Y*>) = <*(Funcs (X,Y))*>

proof end;

theorem :: FUNCT_6:81

for X, Y, Z being set holds Funcs (X,<*Y,Z*>) = <*(Funcs (X,Y)),(Funcs (X,Z))*>

proof end;

theorem :: FUNCT_6:82

for X, Y, Z being set holds Funcs (X,(Y --> Z)) = Y --> (Funcs (X,Z))

proof end;

theorem :: FUNCT_6:83

for X being set
for f being Function holds product (Funcs (X,f)), Funcs (X,(product f)) are_equipotent

proof end;

begin

definition

let f be Function;
canceled;
canceled;
func commute f -> Function-yielding Function equals :: FUNCT_6:def 12
curry' (uncurry f);
coherence

proof end;

end;

:: deftheorem FUNCT_6:def 10 :

:: deftheorem FUNCT_6:def 11 :

:: deftheorem defines commute FUNCT_6:def 12 :

theorem :: FUNCT_6:84

for f being Function
for x being set st x in dom (commute f) holds
(commute f) . x is Function ;

theorem Th85: :: FUNCT_6:85

for A, B, C being set
for f being Function st A <> {} & B <> {} & f in Funcs (A,(Funcs (B,C))) holds
commute f in Funcs (B,(Funcs (A,C)))

proof end;

theorem :: FUNCT_6:86

for A, B, C being set
for f being Function st A <> {} & B <> {} & f in Funcs (A,(Funcs (B,C))) holds
for g, h being Function
for x, y being set st x in A & y in B & f . x = g & (commute f) . y = h holds
( h . x = g . y & dom h = A & dom g = B & rng h c= C & rng g c= C )

proof end;

theorem :: FUNCT_6:87

for A, B, C being set
for f being Function st A <> {} & B <> {} & f in Funcs (A,(Funcs (B,C))) holds
commute (commute f) = f

proof end;

theorem :: FUNCT_6:88

commute {} = {} by FUNCT_5:49, FUNCT_5:50;

theorem :: FUNCT_6:89

for f being Function-yielding Function holds dom (doms f) = dom f

proof end;

theorem :: FUNCT_6:90

for f being Function-yielding Function holds dom (rngs f) = dom f

proof end;