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Theorem upxp 12870
Description: Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
upxp.1  |-  P  =  ( 1st  |`  ( B  X.  C ) )
upxp.2  |-  Q  =  ( 2nd  |`  ( B  X.  C ) )
Assertion
Ref Expression
upxp  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  E! h ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )
Distinct variable groups:    A, h    B, h    C, h    h, F   
h, G    D, h
Allowed substitution hints:    P( h)    Q( h)

Proof of Theorem upxp
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mptexg 5707 . . . 4  |-  ( A  e.  D  ->  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  e.  _V )
2 eueq 2895 . . . 4  |-  ( ( x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  e.  _V  <->  E! h  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
)
31, 2sylib 121 . . 3  |-  ( A  e.  D  ->  E! h  h  =  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) )
433ad2ant1 1007 . 2  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  E! h  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
5 ffn 5334 . . . . . . . 8  |-  ( h : A --> ( B  X.  C )  ->  h  Fn  A )
653ad2ant1 1007 . . . . . . 7  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  ->  h  Fn  A )
76adantl 275 . . . . . 6  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  ->  h  Fn  A
)
8 ffvelrn 5615 . . . . . . . . . . . . 13  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
9 ffvelrn 5615 . . . . . . . . . . . . 13  |-  ( ( G : A --> C  /\  x  e.  A )  ->  ( G `  x
)  e.  C )
10 opelxpi 4633 . . . . . . . . . . . . 13  |-  ( ( ( F `  x
)  e.  B  /\  ( G `  x )  e.  C )  ->  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
118, 9, 10syl2an 287 . . . . . . . . . . . 12  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  ( G : A --> C  /\  x  e.  A ) )  ->  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
1211anandirs 583 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  G : A --> C )  /\  x  e.  A )  ->  <. ( F `  x ) ,  ( G `  x ) >.  e.  ( B  X.  C ) )
1312ralrimiva 2537 . . . . . . . . . 10  |-  ( ( F : A --> B  /\  G : A --> C )  ->  A. x  e.  A  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
14133adant1 1004 . . . . . . . . 9  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  A. x  e.  A  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
15 eqid 2164 . . . . . . . . . 10  |-  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )
1615fmpt 5632 . . . . . . . . 9  |-  ( A. x  e.  A  <. ( F `  x ) ,  ( G `  x ) >.  e.  ( B  X.  C )  <-> 
( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) : A --> ( B  X.  C ) )
1714, 16sylib 121 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) : A --> ( B  X.  C ) )
1817ffnd 5335 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  Fn  A
)
1918adantr 274 . . . . . 6  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  ->  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  Fn  A
)
20 xpss 4709 . . . . . . . . . . 11  |-  ( B  X.  C )  C_  ( _V  X.  _V )
21 ffvelrn 5615 . . . . . . . . . . 11  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( h `  z
)  e.  ( B  X.  C ) )
2220, 21sselid 3138 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( h `  z
)  e.  ( _V 
X.  _V ) )
23223ad2antl1 1148 . . . . . . . . 9  |-  ( ( ( h : A --> ( B  X.  C
)  /\  F  =  ( P  o.  h
)  /\  G  =  ( Q  o.  h
) )  /\  z  e.  A )  ->  (
h `  z )  e.  ( _V  X.  _V ) )
2423adantll 468 . . . . . . . 8  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( h `  z )  e.  ( _V  X.  _V )
)
25 fveq1 5482 . . . . . . . . . . . 12  |-  ( F  =  ( P  o.  h )  ->  ( F `  z )  =  ( ( P  o.  h ) `  z ) )
26 upxp.1 . . . . . . . . . . . . . 14  |-  P  =  ( 1st  |`  ( B  X.  C ) )
2726coeq1i 4760 . . . . . . . . . . . . 13  |-  ( P  o.  h )  =  ( ( 1st  |`  ( B  X.  C ) )  o.  h )
2827fveq1i 5484 . . . . . . . . . . . 12  |-  ( ( P  o.  h ) `
 z )  =  ( ( ( 1st  |`  ( B  X.  C
) )  o.  h
) `  z )
2925, 28eqtrdi 2213 . . . . . . . . . . 11  |-  ( F  =  ( P  o.  h )  ->  ( F `  z )  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  h ) `  z
) )
30293ad2ant2 1008 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  -> 
( F `  z
)  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  h ) `  z ) )
3130ad2antlr 481 . . . . . . . . 9  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( F `  z )  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  h ) `  z ) )
32 simpr1 992 . . . . . . . . . 10  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  ->  h : A --> ( B  X.  C
) )
33 fvco3 5554 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 1st  |`  ( B  X.  C
) )  o.  h
) `  z )  =  ( ( 1st  |`  ( B  X.  C
) ) `  (
h `  z )
) )
3432, 33sylan 281 . . . . . . . . 9  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( (
( 1st  |`  ( B  X.  C ) )  o.  h ) `  z )  =  ( ( 1st  |`  ( B  X.  C ) ) `
 ( h `  z ) ) )
35213ad2antl1 1148 . . . . . . . . . . 11  |-  ( ( ( h : A --> ( B  X.  C
)  /\  F  =  ( P  o.  h
)  /\  G  =  ( Q  o.  h
) )  /\  z  e.  A )  ->  (
h `  z )  e.  ( B  X.  C
) )
3635adantll 468 . . . . . . . . . 10  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( h `  z )  e.  ( B  X.  C ) )
3736fvresd 5508 . . . . . . . . 9  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( ( 1st  |`  ( B  X.  C ) ) `  ( h `  z
) )  =  ( 1st `  ( h `
 z ) ) )
3831, 34, 373eqtrrd 2202 . . . . . . . 8  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( 1st `  ( h `  z
) )  =  ( F `  z ) )
39 fveq1 5482 . . . . . . . . . . . 12  |-  ( G  =  ( Q  o.  h )  ->  ( G `  z )  =  ( ( Q  o.  h ) `  z ) )
40 upxp.2 . . . . . . . . . . . . . 14  |-  Q  =  ( 2nd  |`  ( B  X.  C ) )
4140coeq1i 4760 . . . . . . . . . . . . 13  |-  ( Q  o.  h )  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  h )
4241fveq1i 5484 . . . . . . . . . . . 12  |-  ( ( Q  o.  h ) `
 z )  =  ( ( ( 2nd  |`  ( B  X.  C
) )  o.  h
) `  z )
4339, 42eqtrdi 2213 . . . . . . . . . . 11  |-  ( G  =  ( Q  o.  h )  ->  ( G `  z )  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z
) )
44433ad2ant3 1009 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  -> 
( G `  z
)  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z ) )
4544ad2antlr 481 . . . . . . . . 9  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( G `  z )  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z ) )
46 fvco3 5554 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 2nd  |`  ( B  X.  C
) )  o.  h
) `  z )  =  ( ( 2nd  |`  ( B  X.  C
) ) `  (
h `  z )
) )
4732, 46sylan 281 . . . . . . . . 9  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( (
( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z )  =  ( ( 2nd  |`  ( B  X.  C ) ) `
 ( h `  z ) ) )
4836fvresd 5508 . . . . . . . . 9  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( ( 2nd  |`  ( B  X.  C ) ) `  ( h `  z
) )  =  ( 2nd `  ( h `
 z ) ) )
4945, 47, 483eqtrrd 2202 . . . . . . . 8  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( 2nd `  ( h `  z
) )  =  ( G `  z ) )
50 eqopi 6135 . . . . . . . 8  |-  ( ( ( h `  z
)  e.  ( _V 
X.  _V )  /\  (
( 1st `  (
h `  z )
)  =  ( F `
 z )  /\  ( 2nd `  ( h `
 z ) )  =  ( G `  z ) ) )  ->  ( h `  z )  =  <. ( F `  z ) ,  ( G `  z ) >. )
5124, 38, 49, 50syl12anc 1225 . . . . . . 7  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( h `  z )  =  <. ( F `  z ) ,  ( G `  z ) >. )
52 fveq2 5483 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
53 fveq2 5483 . . . . . . . . 9  |-  ( x  =  z  ->  ( G `  x )  =  ( G `  z ) )
5452, 53opeq12d 3763 . . . . . . . 8  |-  ( x  =  z  ->  <. ( F `  x ) ,  ( G `  x ) >.  =  <. ( F `  z ) ,  ( G `  z ) >. )
55 simpr 109 . . . . . . . 8  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  z  e.  A )
5651, 36eqeltrrd 2242 . . . . . . . 8  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  <. ( F `
 z ) ,  ( G `  z
) >.  e.  ( B  X.  C ) )
5715, 54, 55, 56fvmptd3 5576 . . . . . . 7  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) `  z )  =  <. ( F `  z ) ,  ( G `  z )
>. )
5851, 57eqtr4d 2200 . . . . . 6  |-  ( ( ( ( A  e.  D  /\  F : A
--> B  /\  G : A
--> C )  /\  (
h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  /\  z  e.  A
)  ->  ( h `  z )  =  ( ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) `  z )
)
597, 19, 58eqfnfvd 5583 . . . . 5  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )  ->  h  =  ( x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) )
6059ex 114 . . . 4  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  ->  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
61 ffn 5334 . . . . . . . . 9  |-  ( F : A --> B  ->  F  Fn  A )
62613ad2ant2 1008 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  Fn  A
)
63 fo1st 6120 . . . . . . . . . . 11  |-  1st : _V -onto-> _V
64 fofn 5409 . . . . . . . . . . 11  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
6563, 64ax-mp 5 . . . . . . . . . 10  |-  1st  Fn  _V
66 ssv 3162 . . . . . . . . . 10  |-  ( B  X.  C )  C_  _V
67 fnssres 5298 . . . . . . . . . 10  |-  ( ( 1st  Fn  _V  /\  ( B  X.  C
)  C_  _V )  ->  ( 1st  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
6865, 66, 67mp2an 423 . . . . . . . . 9  |-  ( 1st  |`  ( B  X.  C
) )  Fn  ( B  X.  C )
6917frnd 5344 . . . . . . . . 9  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ran  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )  C_  ( B  X.  C
) )
70 fnco 5293 . . . . . . . . 9  |-  ( ( ( 1st  |`  ( B  X.  C ) )  Fn  ( B  X.  C )  /\  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  Fn  A  /\  ran  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
)  C_  ( B  X.  C ) )  -> 
( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  Fn  A
)
7168, 18, 69, 70mp3an2i 1331 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( ( 1st  |`  ( B  X.  C
) )  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) )  Fn  A
)
72 fvco3 5554 . . . . . . . . . 10  |-  ( ( ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 1st  |`  ( B  X.  C
) )  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) `  z
)  =  ( ( 1st  |`  ( B  X.  C ) ) `  ( ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) `  z ) ) )
7317, 72sylan 281 . . . . . . . . 9  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( (
( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) `  z
)  =  ( ( 1st  |`  ( B  X.  C ) ) `  ( ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) `  z ) ) )
74 simpr 109 . . . . . . . . . . 11  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  z  e.  A )
75 simpl2 990 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  F : A
--> B )
7675, 74ffvelrnd 5618 . . . . . . . . . . . 12  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( F `  z )  e.  B
)
77 simpl3 991 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  G : A
--> C )
7877, 74ffvelrnd 5618 . . . . . . . . . . . 12  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( G `  z )  e.  C
)
7976, 78opelxpd 4634 . . . . . . . . . . 11  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  <. ( F `
 z ) ,  ( G `  z
) >.  e.  ( B  X.  C ) )
8015, 54, 74, 79fvmptd3 5576 . . . . . . . . . 10  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) `  z )  =  <. ( F `  z ) ,  ( G `  z )
>. )
8180fveq2d 5487 . . . . . . . . 9  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( ( 1st  |`  ( B  X.  C ) ) `  ( ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) `  z ) )  =  ( ( 1st  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
) )
82 ffvelrn 5615 . . . . . . . . . . . . . 14  |-  ( ( F : A --> B  /\  z  e.  A )  ->  ( F `  z
)  e.  B )
83 ffvelrn 5615 . . . . . . . . . . . . . 14  |-  ( ( G : A --> C  /\  z  e.  A )  ->  ( G `  z
)  e.  C )
84 opelxpi 4633 . . . . . . . . . . . . . 14  |-  ( ( ( F `  z
)  e.  B  /\  ( G `  z )  e.  C )  ->  <. ( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
) )
8582, 83, 84syl2an 287 . . . . . . . . . . . . 13  |-  ( ( ( F : A --> B  /\  z  e.  A
)  /\  ( G : A --> C  /\  z  e.  A ) )  ->  <. ( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
) )
8685anandirs 583 . . . . . . . . . . . 12  |-  ( ( ( F : A --> B  /\  G : A --> C )  /\  z  e.  A )  ->  <. ( F `  z ) ,  ( G `  z ) >.  e.  ( B  X.  C ) )
87863adantl1 1142 . . . . . . . . . . 11  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  <. ( F `
 z ) ,  ( G `  z
) >.  e.  ( B  X.  C ) )
8887fvresd 5508 . . . . . . . . . 10  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( ( 1st  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( 1st `  <. ( F `  z ) ,  ( G `  z )
>. ) )
89 op1stg 6113 . . . . . . . . . . 11  |-  ( ( ( F `  z
)  e.  B  /\  ( G `  z )  e.  C )  -> 
( 1st `  <. ( F `  z ) ,  ( G `  z ) >. )  =  ( F `  z ) )
9076, 78, 89syl2anc 409 . . . . . . . . . 10  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( 1st ` 
<. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( F `
 z ) )
9188, 90eqtrd 2197 . . . . . . . . 9  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( ( 1st  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( F `
 z ) )
9273, 81, 913eqtrrd 2202 . . . . . . . 8  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( F `  z )  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) `  z
) )
9362, 71, 92eqfnfvd 5583 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  =  ( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) )
9426coeq1i 4760 . . . . . . 7  |-  ( P  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  =  ( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
9593, 94eqtr4di 2215 . . . . . 6  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  =  ( P  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
96 ffn 5334 . . . . . . . . 9  |-  ( G : A --> C  ->  G  Fn  A )
97963ad2ant3 1009 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  Fn  A
)
98 fo2nd 6121 . . . . . . . . . . 11  |-  2nd : _V -onto-> _V
99 fofn 5409 . . . . . . . . . . 11  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
10098, 99ax-mp 5 . . . . . . . . . 10  |-  2nd  Fn  _V
101 fnssres 5298 . . . . . . . . . 10  |-  ( ( 2nd  Fn  _V  /\  ( B  X.  C
)  C_  _V )  ->  ( 2nd  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
102100, 66, 101mp2an 423 . . . . . . . . 9  |-  ( 2nd  |`  ( B  X.  C
) )  Fn  ( B  X.  C )
103 fnco 5293 . . . . . . . . 9  |-  ( ( ( 2nd  |`  ( B  X.  C ) )  Fn  ( B  X.  C )  /\  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  Fn  A  /\  ran  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
)  C_  ( B  X.  C ) )  -> 
( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  Fn  A
)
104102, 18, 69, 103mp3an2i 1331 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( ( 2nd  |`  ( B  X.  C
) )  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) )  Fn  A
)
105 fvco3 5554 . . . . . . . . . 10  |-  ( ( ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 2nd  |`  ( B  X.  C
) )  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) `  z
)  =  ( ( 2nd  |`  ( B  X.  C ) ) `  ( ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) `  z ) ) )
10617, 105sylan 281 . . . . . . . . 9  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( (
( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) `  z
)  =  ( ( 2nd  |`  ( B  X.  C ) ) `  ( ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) `  z ) ) )
10780fveq2d 5487 . . . . . . . . 9  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( ( 2nd  |`  ( B  X.  C ) ) `  ( ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) `  z ) )  =  ( ( 2nd  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
) )
10887fvresd 5508 . . . . . . . . . 10  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( ( 2nd  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( 2nd `  <. ( F `  z ) ,  ( G `  z )
>. ) )
109 op2ndg 6114 . . . . . . . . . . 11  |-  ( ( ( F `  z
)  e.  B  /\  ( G `  z )  e.  C )  -> 
( 2nd `  <. ( F `  z ) ,  ( G `  z ) >. )  =  ( G `  z ) )
11076, 78, 109syl2anc 409 . . . . . . . . . 10  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( 2nd ` 
<. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( G `
 z ) )
111108, 110eqtrd 2197 . . . . . . . . 9  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( ( 2nd  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( G `
 z ) )
112106, 107, 1113eqtrrd 2202 . . . . . . . 8  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( G `  z )  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) `  z
) )
11397, 104, 112eqfnfvd 5583 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) )
11440coeq1i 4760 . . . . . . 7  |-  ( Q  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
115113, 114eqtr4di 2215 . . . . . 6  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  =  ( Q  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
11617, 95, 1153jca 1166 . . . . 5  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. ) : A --> ( B  X.  C )  /\  F  =  ( P  o.  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) )  /\  G  =  ( Q  o.  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) ) ) )
117 feq1 5317 . . . . . 6  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( h : A --> ( B  X.  C )  <->  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. ) : A --> ( B  X.  C ) ) )
118 coeq2 4759 . . . . . . 7  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( P  o.  h )  =  ( P  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) )
119118eqeq2d 2176 . . . . . 6  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( F  =  ( P  o.  h )  <->  F  =  ( P  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) ) )
120 coeq2 4759 . . . . . . 7  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( Q  o.  h )  =  ( Q  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) )
121120eqeq2d 2176 . . . . . 6  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( G  =  ( Q  o.  h )  <->  G  =  ( Q  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) ) )
122117, 119, 1213anbi123d 1301 . . . . 5  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  <->  ( (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) : A --> ( B  X.  C )  /\  F  =  ( P  o.  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) )  /\  G  =  ( Q  o.  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) ) ) ) )
123116, 122syl5ibrcom 156 . . . 4  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( h  =  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
)  ->  ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) ) )
12460, 123impbid 128 . . 3  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  <->  h  =  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) ) )
125124eubidv 2021 . 2  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( E! h
( h : A --> ( B  X.  C
)  /\  F  =  ( P  o.  h
)  /\  G  =  ( Q  o.  h
) )  <->  E! h  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
1264, 125mpbird 166 1  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  E! h ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 967    = wceq 1342   E!weu 2013    e. wcel 2135   A.wral 2442   _Vcvv 2724    C_ wss 3114   <.cop 3576    |-> cmpt 4040    X. cxp 4599   ran crn 4602    |` cres 4603    o. ccom 4605    Fn wfn 5180   -->wf 5181   -onto->wfo 5183   ` cfv 5185   1stc1st 6101   2ndc2nd 6102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4094  ax-sep 4097  ax-pow 4150  ax-pr 4184  ax-un 4408
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2726  df-sbc 2950  df-csb 3044  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-iun 3865  df-br 3980  df-opab 4041  df-mpt 4042  df-id 4268  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-f 5189  df-f1 5190  df-fo 5191  df-f1o 5192  df-fv 5193  df-1st 6103  df-2nd 6104
This theorem is referenced by:  uptx  12872  txcn  12873
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