:: Cartesian Product of Functions
:: by Grzegorz Bancerek
::
:: Received September 30, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users

theorem Th1: :: FUNCT_6:1

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

proof end;

theorem Th2: :: FUNCT_6:2

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

proof end;

theorem Th3: :: FUNCT_6:3

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

proof end;

theorem Th4: :: FUNCT_6:4

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

proof end;

theorem :: FUNCT_6:5

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

proof end;

theorem :: FUNCT_6:6

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

proof end;

theorem Th7: :: FUNCT_6:7

for x being object
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 Th8: :: FUNCT_6:8

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_6:9

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 Th10: :: FUNCT_6:10

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 Th11: :: FUNCT_6:11

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_6:12

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_6:13

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_6:14

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 Th15: :: FUNCT_6:15

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_6:16

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_6:17

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;

definition

end;

:: deftheorem FUNCT_6:def 1 :
canceled;

::$CD

theorem :: FUNCT_6:18

canceled;

::$CD

theorem :: FUNCT_6:19

canceled;

::$CD

theorem :: FUNCT_6:20

canceled;

::$CD

theorem :: FUNCT_6:21

canceled;

::$CT 4

definition

let f be Function-yielding Function;

func doms f -> Function means :Def1: :: FUNCT_6:def 2
( dom it = dom f & ( for x being object st x in dom f holds
it . x = proj1 (f . x) ) );

existence
ex b₁ being Function st
( dom b₁ = dom f & ( for x being object st x in dom f holds
b₁ . x = proj1 (f . x) ) )

proof end;

uniqueness
for b₁, b₂ being Function st dom b₁ = dom f & ( for x being object st x in dom f holds
b₁ . x = proj1 (f . x) ) & dom b₂ = dom f & ( for x being object st x in dom f holds
b₂ . x = proj1 (f . x) ) holds
b₁ = b₂

proof end;

func rngs f -> Function means :Def2: :: FUNCT_6:def 3
( dom it = dom f & ( for x being object st x in dom f holds
it . x = proj2 (f . x) ) );

existence
ex b₁ being Function st
( dom b₁ = dom f & ( for x being object st x in dom f holds
b₁ . x = proj2 (f . x) ) )

proof end;

uniqueness
for b₁, b₂ being Function st dom b₁ = dom f & ( for x being object st x in dom f holds
b₁ . x = proj2 (f . x) ) & dom b₂ = dom f & ( for x being object st x in dom f holds
b₂ . x = proj2 (f . x) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines doms FUNCT_6:def 2 :
for f being Function-yielding Function
for b₂ being Function holds
( b₂ = doms f iff ( dom b₂ = dom f & ( for x being object st x in dom f holds
b₂ . x = proj1 (f . x) ) ) );

:: deftheorem Def2 defines rngs FUNCT_6:def 3 :
for f being Function-yielding Function
for b₂ being Function holds
( b₂ = rngs f iff ( dom b₂ = dom f & ( for x being object st x in dom f holds
b₂ . x = proj2 (f . x) ) ) );

definition

let f be Function;

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

correctness
coherence
meet (rng f) is set ;
;

end;

:: deftheorem defines meet FUNCT_6:def 4 :
for f being Function holds meet f = meet (rng f);

theorem Th18: :: FUNCT_6:22

for x being object
for g being Function
for f being Function-yielding 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 ) by Def1, Def2;

theorem :: FUNCT_6:23

( doms {} = {} & rngs {} = {} ) by Def1, Def2, RELAT_1:38;

theorem Th20: :: FUNCT_6:24

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

proof end;

theorem Th21: :: FUNCT_6:25

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

proof end;

theorem :: FUNCT_6:26

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

proof end;

theorem Th23: :: FUNCT_6:27

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 object ;

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

correctness
coherence
(uncurry f) . (x,y) is set ;
;

end;

:: deftheorem defines .. FUNCT_6:def 5 :
for f being Function
for x, y being object holds f .. (x,y) = (uncurry f) . (x,y);

theorem :: FUNCT_6:28

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

proof end;

definition

let f be Function-yielding Function;

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

correctness
coherence
curry ((uncurry' f) | [:(meet (doms f)),(dom f):]) is Function;
;

end;

:: deftheorem defines <: FUNCT_6:def 6 :
for f being Function-yielding Function holds <:f:> = curry ((uncurry' f) | [:(meet (doms f)),(dom f):]);

theorem Th25: :: FUNCT_6:29

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

proof end;

registration

let f be Function-yielding Function;

cluster <:f:> -> Function-yielding ;

coherence
<:f:> is Function-yielding ;

end;

theorem :: FUNCT_6:30

canceled;

::$CT

theorem Th26: :: FUNCT_6:31

for x being object
for g being Function
for f being Function-yielding Function st x in dom <:f:> & g = <:f:> . x holds
( dom g = dom f & ( for y being object st y in dom g holds
( [y,x] in dom (uncurry f) & g . y = (uncurry f) . (y,x) ) ) )

proof end;

theorem Th27: :: FUNCT_6:32

for x being object
for f being Function-yielding 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_6:33

for g being Function
for x being object
for f being Function-yielding 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 Th29: :: FUNCT_6:34

for x, y being object
for g, h being Function
for f being Function-yielding 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_6:35

for x, y being object
for f being Function-yielding 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-yielding Function;

func Frege f -> Function means :Def6: :: 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 = dom f & ( for x being object st x in dom h holds
h . x = (uncurry f) . (x,(g . x)) ) ) ) );

existence
ex b₁ being Function st
( dom b₁ = product (doms f) & ( for g being Function st g in product (doms f) holds
ex h being Function st
( b₁ . g = h & dom h = dom f & ( for x being object st x in dom h holds
h . x = (uncurry f) . (x,(g . x)) ) ) ) )

proof end;

uniqueness
for b₁, b₂ being Function st dom b₁ = product (doms f) & ( for g being Function st g in product (doms f) holds
ex h being Function st
( b₁ . g = h & dom h = dom f & ( for x being object st x in dom h holds
h . x = (uncurry f) . (x,(g . x)) ) ) ) & dom b₂ = product (doms f) & ( for g being Function st g in product (doms f) holds
ex h being Function st
( b₂ . g = h & dom h = dom f & ( for x being object st x in dom h holds
h . x = (uncurry f) . (x,(g . x)) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines Frege FUNCT_6:def 7 :
for f being Function-yielding Function
for b₂ being Function holds
( b₂ = Frege f iff ( dom b₂ = product (doms f) & ( for g being Function st g in product (doms f) holds
ex h being Function st
( b₂ . g = h & dom h = dom f & ( for x being object st x in dom h holds
h . x = (uncurry f) . (x,(g . x)) ) ) ) ) );

theorem :: FUNCT_6:36

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

proof end;

Lm1: for f being Function-yielding Function holds rng (Frege f) c= product (rngs f)

proof end;

theorem :: FUNCT_6:37

for x being object
for g, h, h9 being Function
for f being Function-yielding 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;

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

proof end;

theorem Th33: :: FUNCT_6:38

for f being Function-yielding Function holds rng (Frege f) = product (rngs f) by Lm1, Lm2;

theorem Th34: :: FUNCT_6:39

for f being Function-yielding 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;

theorem :: FUNCT_6:40

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

proof end;

theorem :: FUNCT_6:41

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

proof end;

theorem :: FUNCT_6:42

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 Th38: :: FUNCT_6:43

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

proof end;

theorem Th39: :: FUNCT_6:44

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

proof end;

theorem :: FUNCT_6:45

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;

definition

let f be Function;
let X be set ;

func Funcs (f,X) -> Function means :Def7: :: FUNCT_6:def 8
( dom it = dom f & ( for x being object st x in dom f holds
it . x = Funcs ((f . x),X) ) );

existence
ex b₁ being Function st
( dom b₁ = dom f & ( for x being object st x in dom f holds
b₁ . x = Funcs ((f . x),X) ) )

proof end;

uniqueness
for b₁, b₂ being Function st dom b₁ = dom f & ( for x being object st x in dom f holds
b₁ . x = Funcs ((f . x),X) ) & dom b₂ = dom f & ( for x being object st x in dom f holds
b₂ . x = Funcs ((f . x),X) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines Funcs FUNCT_6:def 8 :
for f being Function
for X being set
for b₃ being Function holds
( b₃ = Funcs (f,X) iff ( dom b₃ = dom f & ( for x being object st x in dom f holds
b₃ . x = Funcs ((f . x),X) ) ) );

theorem :: FUNCT_6:46

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

proof end;

theorem :: FUNCT_6:47

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

proof end;

theorem :: FUNCT_6:48

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

proof end;

Lm3: for f, g being Function
for x, y, z being object st [x,y] in dom f & g = (curry f) . x & z in dom g holds
[x,z] in dom f

proof end;

theorem :: FUNCT_6:49

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 :Def8: :: FUNCT_6:def 9
( dom it = dom f & ( for x being object st x in dom f holds
it . x = Funcs (X,(f . x)) ) );

existence
ex b₁ being Function st
( dom b₁ = dom f & ( for x being object st x in dom f holds
b₁ . x = Funcs (X,(f . x)) ) )

proof end;

uniqueness
for b₁, b₂ being Function st dom b₁ = dom f & ( for x being object st x in dom f holds
b₁ . x = Funcs (X,(f . x)) ) & dom b₂ = dom f & ( for x being object st x in dom f holds
b₂ . x = Funcs (X,(f . x)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines Funcs FUNCT_6:def 9 :
for X being set
for f, b₃ being Function holds
( b₃ = Funcs (X,f) iff ( dom b₃ = dom f & ( for x being object st x in dom f holds
b₃ . x = Funcs (X,(f . x)) ) ) );

Lm4: 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 Th45: :: FUNCT_6:50

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

proof end;

theorem Th46: :: FUNCT_6:51

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

proof end;

theorem :: FUNCT_6:52

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

proof end;

theorem :: FUNCT_6:53

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

proof end;

:: from PRALG_2

definition

let f be Function;

func commute f -> Function-yielding Function equals :: FUNCT_6:def 10
curry' (uncurry f);

coherence
curry' (uncurry f) is Function-yielding Function ;

end;

:: deftheorem defines commute FUNCT_6:def 10 :
for f being Function holds commute f = curry' (uncurry f);

theorem :: FUNCT_6:54

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

theorem Th50: :: FUNCT_6:55

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_6:56

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_6:57

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_6:58

commute {} = {} by FUNCT_5:42, FUNCT_5:43;

:: from EXTENS_1

theorem :: FUNCT_6:59

for f being Function-yielding Function holds dom (doms f) = dom f by Def1;

theorem :: FUNCT_6:60

for f being Function-yielding Function holds dom (rngs f) = dom f by Def2;