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Theorem upxp 20250
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 6143 . . . 4  |-  ( A  e.  D  ->  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  e.  _V )
2 eueq 3271 . . . 4  |-  ( ( x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  e.  _V  <->  E! h  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
)
31, 2sylib 196 . . 3  |-  ( A  e.  D  ->  E! h  h  =  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) )
433ad2ant1 1017 . 2  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  E! h  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
5 ffn 5737 . . . . . . . 8  |-  ( h : A --> ( B  X.  C )  ->  h  Fn  A )
653ad2ant1 1017 . . . . . . 7  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  ->  h  Fn  A )
76adantl 466 . . . . . 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 6030 . . . . . . . . . . . . 13  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
9 ffvelrn 6030 . . . . . . . . . . . . 13  |-  ( ( G : A --> C  /\  x  e.  A )  ->  ( G `  x
)  e.  C )
10 opelxpi 5040 . . . . . . . . . . . . 13  |-  ( ( ( F `  x
)  e.  B  /\  ( G `  x )  e.  C )  ->  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
118, 9, 10syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  ( G : A --> C  /\  x  e.  A ) )  ->  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
1211anandirs 831 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  G : A --> C )  /\  x  e.  A )  ->  <. ( F `  x ) ,  ( G `  x ) >.  e.  ( B  X.  C ) )
1312ralrimiva 2871 . . . . . . . . . 10  |-  ( ( F : A --> B  /\  G : A --> C )  ->  A. x  e.  A  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
14133adant1 1014 . . . . . . . . 9  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  A. x  e.  A  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
15 eqid 2457 . . . . . . . . . 10  |-  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )
1615fmpt 6053 . . . . . . . . 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 196 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) : A --> ( B  X.  C ) )
18 ffn 5737 . . . . . . . 8  |-  ( ( x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) : A --> ( B  X.  C )  -> 
( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
)  Fn  A )
1917, 18syl 16 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  Fn  A
)
2019adantr 465 . . . . . 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
)
21 xpss 5118 . . . . . . . . . . 11  |-  ( B  X.  C )  C_  ( _V  X.  _V )
22 ffvelrn 6030 . . . . . . . . . . 11  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( h `  z
)  e.  ( B  X.  C ) )
2321, 22sseldi 3497 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( h `  z
)  e.  ( _V 
X.  _V ) )
24233ad2antl1 1158 . . . . . . . . 9  |-  ( ( ( h : A --> ( B  X.  C
)  /\  F  =  ( P  o.  h
)  /\  G  =  ( Q  o.  h
) )  /\  z  e.  A )  ->  (
h `  z )  e.  ( _V  X.  _V ) )
2524adantll 713 . . . . . . . 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 )
)
26 fveq1 5871 . . . . . . . . . . . 12  |-  ( F  =  ( P  o.  h )  ->  ( F `  z )  =  ( ( P  o.  h ) `  z ) )
27 upxp.1 . . . . . . . . . . . . . 14  |-  P  =  ( 1st  |`  ( B  X.  C ) )
2827coeq1i 5172 . . . . . . . . . . . . 13  |-  ( P  o.  h )  =  ( ( 1st  |`  ( B  X.  C ) )  o.  h )
2928fveq1i 5873 . . . . . . . . . . . 12  |-  ( ( P  o.  h ) `
 z )  =  ( ( ( 1st  |`  ( B  X.  C
) )  o.  h
) `  z )
3026, 29syl6eq 2514 . . . . . . . . . . 11  |-  ( F  =  ( P  o.  h )  ->  ( F `  z )  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  h ) `  z
) )
31303ad2ant2 1018 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  -> 
( F `  z
)  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  h ) `  z ) )
3231ad2antlr 726 . . . . . . . . 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 ) )
33 simpr1 1002 . . . . . . . . . 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
) )
34 fvco3 5950 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 1st  |`  ( B  X.  C
) )  o.  h
) `  z )  =  ( ( 1st  |`  ( B  X.  C
) ) `  (
h `  z )
) )
3533, 34sylan 471 . . . . . . . . 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 ) ) )
36223ad2antl1 1158 . . . . . . . . . . 11  |-  ( ( ( h : A --> ( B  X.  C
)  /\  F  =  ( P  o.  h
)  /\  G  =  ( Q  o.  h
) )  /\  z  e.  A )  ->  (
h `  z )  e.  ( B  X.  C
) )
3736adantll 713 . . . . . . . . . 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 ) )
38 fvres 5886 . . . . . . . . . 10  |-  ( ( h `  z )  e.  ( B  X.  C )  ->  (
( 1st  |`  ( B  X.  C ) ) `
 ( h `  z ) )  =  ( 1st `  (
h `  z )
) )
3937, 38syl 16 . . . . . . . . 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 ) ) )
4032, 35, 393eqtrrd 2503 . . . . . . . 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 ) )
41 fveq1 5871 . . . . . . . . . . . 12  |-  ( G  =  ( Q  o.  h )  ->  ( G `  z )  =  ( ( Q  o.  h ) `  z ) )
42 upxp.2 . . . . . . . . . . . . . 14  |-  Q  =  ( 2nd  |`  ( B  X.  C ) )
4342coeq1i 5172 . . . . . . . . . . . . 13  |-  ( Q  o.  h )  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  h )
4443fveq1i 5873 . . . . . . . . . . . 12  |-  ( ( Q  o.  h ) `
 z )  =  ( ( ( 2nd  |`  ( B  X.  C
) )  o.  h
) `  z )
4541, 44syl6eq 2514 . . . . . . . . . . 11  |-  ( G  =  ( Q  o.  h )  ->  ( G `  z )  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z
) )
46453ad2ant3 1019 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  -> 
( G `  z
)  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z ) )
4746ad2antlr 726 . . . . . . . . 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 ) )
48 fvco3 5950 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 2nd  |`  ( B  X.  C
) )  o.  h
) `  z )  =  ( ( 2nd  |`  ( B  X.  C
) ) `  (
h `  z )
) )
4933, 48sylan 471 . . . . . . . . 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 ) ) )
50 fvres 5886 . . . . . . . . . 10  |-  ( ( h `  z )  e.  ( B  X.  C )  ->  (
( 2nd  |`  ( B  X.  C ) ) `
 ( h `  z ) )  =  ( 2nd `  (
h `  z )
) )
5137, 50syl 16 . . . . . . . . 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 ) ) )
5247, 49, 513eqtrrd 2503 . . . . . . . 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 ) )
53 eqopi 6833 . . . . . . . 8  |-  ( ( ( h `  z
)  e.  ( _V 
X.  _V )  /\  (
( 1st `  (
h `  z )
)  =  ( F `
 z )  /\  ( 2nd `  ( h `
 z ) )  =  ( G `  z ) ) )  ->  ( h `  z )  =  <. ( F `  z ) ,  ( G `  z ) >. )
5425, 40, 52, 53syl12anc 1226 . . . . . . 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 ) >. )
55 fveq2 5872 . . . . . . . . . 10  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
56 fveq2 5872 . . . . . . . . . 10  |-  ( x  =  z  ->  ( G `  x )  =  ( G `  z ) )
5755, 56opeq12d 4227 . . . . . . . . 9  |-  ( x  =  z  ->  <. ( F `  x ) ,  ( G `  x ) >.  =  <. ( F `  z ) ,  ( G `  z ) >. )
58 opex 4720 . . . . . . . . 9  |-  <. ( F `  z ) ,  ( G `  z ) >.  e.  _V
5957, 15, 58fvmpt 5956 . . . . . . . 8  |-  ( z  e.  A  ->  (
( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) `  z )  =  <. ( F `  z ) ,  ( G `  z )
>. )
6059adantl 466 . . . . . . 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 )
>. )
6154, 60eqtr4d 2501 . . . . . 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 )
)
627, 20, 61eqfnfvd 5985 . . . . 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 ) >.
) )
6362ex 434 . . . 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 ) >. )
) )
64 ffn 5737 . . . . . . . . 9  |-  ( F : A --> B  ->  F  Fn  A )
65643ad2ant2 1018 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  Fn  A
)
66 fo1st 6819 . . . . . . . . . . . 12  |-  1st : _V -onto-> _V
67 fofn 5803 . . . . . . . . . . . 12  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
6866, 67ax-mp 5 . . . . . . . . . . 11  |-  1st  Fn  _V
69 ssv 3519 . . . . . . . . . . 11  |-  ( B  X.  C )  C_  _V
70 fnssres 5700 . . . . . . . . . . 11  |-  ( ( 1st  Fn  _V  /\  ( B  X.  C
)  C_  _V )  ->  ( 1st  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
7168, 69, 70mp2an 672 . . . . . . . . . 10  |-  ( 1st  |`  ( B  X.  C
) )  Fn  ( B  X.  C )
7271a1i 11 . . . . . . . . 9  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( 1st  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
73 frn 5743 . . . . . . . . . 10  |-  ( ( x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) : A --> ( B  X.  C )  ->  ran  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
)  C_  ( B  X.  C ) )
7417, 73syl 16 . . . . . . . . 9  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ran  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )  C_  ( B  X.  C
) )
75 fnco 5695 . . . . . . . . 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
)
7672, 19, 74, 75syl3anc 1228 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( ( 1st  |`  ( B  X.  C
) )  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) )  Fn  A
)
77 fvco3 5950 . . . . . . . . . 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 ) ) )
7817, 77sylan 471 . . . . . . . . 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 ) ) )
7959adantl 466 . . . . . . . . . 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 )
>. )
8079fveq2d 5876 . . . . . . . . 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 ) >.
) )
81 ffvelrn 6030 . . . . . . . . . . . . . 14  |-  ( ( F : A --> B  /\  z  e.  A )  ->  ( F `  z
)  e.  B )
82 ffvelrn 6030 . . . . . . . . . . . . . 14  |-  ( ( G : A --> C  /\  z  e.  A )  ->  ( G `  z
)  e.  C )
83 opelxpi 5040 . . . . . . . . . . . . . 14  |-  ( ( ( F `  z
)  e.  B  /\  ( G `  z )  e.  C )  ->  <. ( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
) )
8481, 82, 83syl2an 477 . . . . . . . . . . . . 13  |-  ( ( ( F : A --> B  /\  z  e.  A
)  /\  ( G : A --> C  /\  z  e.  A ) )  ->  <. ( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
) )
8584anandirs 831 . . . . . . . . . . . 12  |-  ( ( ( F : A --> B  /\  G : A --> C )  /\  z  e.  A )  ->  <. ( F `  z ) ,  ( G `  z ) >.  e.  ( B  X.  C ) )
86853adantl1 1152 . . . . . . . . . . 11  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  <. ( F `
 z ) ,  ( G `  z
) >.  e.  ( B  X.  C ) )
87 fvres 5886 . . . . . . . . . . 11  |-  ( <.
( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
)  ->  ( ( 1st  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( 1st `  <. ( F `  z ) ,  ( G `  z )
>. ) )
8886, 87syl 16 . . . . . . . . . 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 fvex 5882 . . . . . . . . . . 11  |-  ( F `
 z )  e. 
_V
90 fvex 5882 . . . . . . . . . . 11  |-  ( G `
 z )  e. 
_V
9189, 90op1st 6807 . . . . . . . . . 10  |-  ( 1st `  <. ( F `  z ) ,  ( G `  z )
>. )  =  ( F `  z )
9288, 91syl6eq 2514 . . . . . . . . 9  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( ( 1st  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( F `
 z ) )
9378, 80, 923eqtrrd 2503 . . . . . . . 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
) )
9465, 76, 93eqfnfvd 5985 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  =  ( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) )
9527coeq1i 5172 . . . . . . 7  |-  ( P  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  =  ( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
9694, 95syl6eqr 2516 . . . . . 6  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  =  ( P  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
97 ffn 5737 . . . . . . . . 9  |-  ( G : A --> C  ->  G  Fn  A )
98973ad2ant3 1019 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  Fn  A
)
99 fo2nd 6820 . . . . . . . . . . . 12  |-  2nd : _V -onto-> _V
100 fofn 5803 . . . . . . . . . . . 12  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
10199, 100ax-mp 5 . . . . . . . . . . 11  |-  2nd  Fn  _V
102 fnssres 5700 . . . . . . . . . . 11  |-  ( ( 2nd  Fn  _V  /\  ( B  X.  C
)  C_  _V )  ->  ( 2nd  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
103101, 69, 102mp2an 672 . . . . . . . . . 10  |-  ( 2nd  |`  ( B  X.  C
) )  Fn  ( B  X.  C )
104103a1i 11 . . . . . . . . 9  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( 2nd  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
105 fnco 5695 . . . . . . . . 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
)
106104, 19, 74, 105syl3anc 1228 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( ( 2nd  |`  ( B  X.  C
) )  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) )  Fn  A
)
107 fvco3 5950 . . . . . . . . . 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 ) ) )
10817, 107sylan 471 . . . . . . . . 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 ) ) )
10979fveq2d 5876 . . . . . . . . 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 ) >.
) )
110 fvres 5886 . . . . . . . . . . 11  |-  ( <.
( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
)  ->  ( ( 2nd  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( 2nd `  <. ( F `  z ) ,  ( G `  z )
>. ) )
11186, 110syl 16 . . . . . . . . . 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 )
>. ) )
11289, 90op2nd 6808 . . . . . . . . . 10  |-  ( 2nd `  <. ( F `  z ) ,  ( G `  z )
>. )  =  ( G `  z )
113111, 112syl6eq 2514 . . . . . . . . 9  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  ( ( 2nd  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( G `
 z ) )
114108, 109, 1133eqtrrd 2503 . . . . . . . 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
) )
11598, 106, 114eqfnfvd 5985 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) )
11642coeq1i 5172 . . . . . . 7  |-  ( Q  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
117115, 116syl6eqr 2516 . . . . . 6  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  =  ( Q  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
11817, 96, 1173jca 1176 . . . . 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 ) >.
) ) ) )
119 feq1 5719 . . . . . 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 ) ) )
120 coeq2 5171 . . . . . . 7  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( P  o.  h )  =  ( P  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) )
121120eqeq2d 2471 . . . . . 6  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( F  =  ( P  o.  h )  <->  F  =  ( P  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) ) )
122 coeq2 5171 . . . . . . 7  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( Q  o.  h )  =  ( Q  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) )
123122eqeq2d 2471 . . . . . 6  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( G  =  ( Q  o.  h )  <->  G  =  ( Q  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) ) )
124119, 121, 1233anbi123d 1299 . . . . 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 ) >.
) ) ) ) )
125118, 124syl5ibrcom 222 . . . 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 ) ) ) )
12663, 125impbid 191 . . 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 ) >.
) ) )
127126eubidv 2305 . 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 ) >. )
) )
1284, 127mpbird 232 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   E!weu 2283   A.wral 2807   _Vcvv 3109    C_ wss 3471   <.cop 4038    |-> cmpt 4515    X. cxp 5006   ran crn 5009    |` cres 5010    o. ccom 5012    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1st 6799  df-2nd 6800
This theorem is referenced by:  uptx  20252  txcn  20253
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