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Theorem upxp 17657
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 5967 . . . 4  |-  ( A  e.  D  ->  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  e.  _V )
2 eueq 3108 . . . 4  |-  ( ( x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  e.  _V  <->  E! h  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
)
31, 2sylib 190 . . 3  |-  ( A  e.  D  ->  E! h  h  =  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) )
433ad2ant1 979 . 2  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  E! h  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
5 ffn 5593 . . . . . . . 8  |-  ( h : A --> ( B  X.  C )  ->  h  Fn  A )
653ad2ant1 979 . . . . . . 7  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  ->  h  Fn  A )
76adantl 454 . . . . . 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 5870 . . . . . . . . . . . . 13  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
9 ffvelrn 5870 . . . . . . . . . . . . 13  |-  ( ( G : A --> C  /\  x  e.  A )  ->  ( G `  x
)  e.  C )
10 opelxpi 4912 . . . . . . . . . . . . 13  |-  ( ( ( F `  x
)  e.  B  /\  ( G `  x )  e.  C )  ->  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
118, 9, 10syl2an 465 . . . . . . . . . . . 12  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  ( G : A --> C  /\  x  e.  A ) )  ->  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
1211anandirs 806 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  G : A --> C )  /\  x  e.  A )  ->  <. ( F `  x ) ,  ( G `  x ) >.  e.  ( B  X.  C ) )
1312ralrimiva 2791 . . . . . . . . . 10  |-  ( ( F : A --> B  /\  G : A --> C )  ->  A. x  e.  A  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
14133adant1 976 . . . . . . . . 9  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  A. x  e.  A  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
15 eqid 2438 . . . . . . . . . 10  |-  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )
1615fmpt 5892 . . . . . . . . 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 190 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) : A --> ( B  X.  C ) )
18 ffn 5593 . . . . . . . 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 453 . . . . . 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 4984 . . . . . . . . . . 11  |-  ( B  X.  C )  C_  ( _V  X.  _V )
22 ffvelrn 5870 . . . . . . . . . . 11  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( h `  z
)  e.  ( B  X.  C ) )
2321, 22sseldi 3348 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( h `  z
)  e.  ( _V 
X.  _V ) )
24233ad2antl1 1120 . . . . . . . . 9  |-  ( ( ( h : A --> ( B  X.  C
)  /\  F  =  ( P  o.  h
)  /\  G  =  ( Q  o.  h
) )  /\  z  e.  A )  ->  (
h `  z )  e.  ( _V  X.  _V ) )
2524adantll 696 . . . . . . . 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 5729 . . . . . . . . . . . 12  |-  ( F  =  ( P  o.  h )  ->  ( F `  z )  =  ( ( P  o.  h ) `  z ) )
27 upxp.1 . . . . . . . . . . . . . 14  |-  P  =  ( 1st  |`  ( B  X.  C ) )
2827coeq1i 5034 . . . . . . . . . . . . 13  |-  ( P  o.  h )  =  ( ( 1st  |`  ( B  X.  C ) )  o.  h )
2928fveq1i 5731 . . . . . . . . . . . 12  |-  ( ( P  o.  h ) `
 z )  =  ( ( ( 1st  |`  ( B  X.  C
) )  o.  h
) `  z )
3026, 29syl6eq 2486 . . . . . . . . . . 11  |-  ( F  =  ( P  o.  h )  ->  ( F `  z )  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  h ) `  z
) )
31303ad2ant2 980 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  -> 
( F `  z
)  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  h ) `  z ) )
3231ad2antlr 709 . . . . . . . . 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 964 . . . . . . . . . 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 5802 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 1st  |`  ( B  X.  C
) )  o.  h
) `  z )  =  ( ( 1st  |`  ( B  X.  C
) ) `  (
h `  z )
) )
3533, 34sylan 459 . . . . . . . . 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 1120 . . . . . . . . . . 11  |-  ( ( ( h : A --> ( B  X.  C
)  /\  F  =  ( P  o.  h
)  /\  G  =  ( Q  o.  h
) )  /\  z  e.  A )  ->  (
h `  z )  e.  ( B  X.  C
) )
3736adantll 696 . . . . . . . . . 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 5747 . . . . . . . . . 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 2475 . . . . . . . 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 5729 . . . . . . . . . . . 12  |-  ( G  =  ( Q  o.  h )  ->  ( G `  z )  =  ( ( Q  o.  h ) `  z ) )
42 upxp.2 . . . . . . . . . . . . . 14  |-  Q  =  ( 2nd  |`  ( B  X.  C ) )
4342coeq1i 5034 . . . . . . . . . . . . 13  |-  ( Q  o.  h )  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  h )
4443fveq1i 5731 . . . . . . . . . . . 12  |-  ( ( Q  o.  h ) `
 z )  =  ( ( ( 2nd  |`  ( B  X.  C
) )  o.  h
) `  z )
4541, 44syl6eq 2486 . . . . . . . . . . 11  |-  ( G  =  ( Q  o.  h )  ->  ( G `  z )  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z
) )
46453ad2ant3 981 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  -> 
( G `  z
)  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z ) )
4746ad2antlr 709 . . . . . . . . 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 5802 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 2nd  |`  ( B  X.  C
) )  o.  h
) `  z )  =  ( ( 2nd  |`  ( B  X.  C
) ) `  (
h `  z )
) )
4933, 48sylan 459 . . . . . . . . 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 5747 . . . . . . . . . 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 2475 . . . . . . . 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 6385 . . . . . . . 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 1183 . . . . . . 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 5730 . . . . . . . . . 10  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
56 fveq2 5730 . . . . . . . . . 10  |-  ( x  =  z  ->  ( G `  x )  =  ( G `  z ) )
5755, 56opeq12d 3994 . . . . . . . . 9  |-  ( x  =  z  ->  <. ( F `  x ) ,  ( G `  x ) >.  =  <. ( F `  z ) ,  ( G `  z ) >. )
58 opex 4429 . . . . . . . . 9  |-  <. ( F `  z ) ,  ( G `  z ) >.  e.  _V
5957, 15, 58fvmpt 5808 . . . . . . . 8  |-  ( z  e.  A  ->  (
( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) `  z )  =  <. ( F `  z ) ,  ( G `  z )
>. )
6059adantl 454 . . . . . . 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 2473 . . . . . 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 5832 . . . . 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 425 . . . 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 5593 . . . . . . . . 9  |-  ( F : A --> B  ->  F  Fn  A )
65643ad2ant2 980 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  Fn  A
)
66 fo1st 6368 . . . . . . . . . . . 12  |-  1st : _V -onto-> _V
67 fofn 5657 . . . . . . . . . . . 12  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
6866, 67ax-mp 8 . . . . . . . . . . 11  |-  1st  Fn  _V
69 ssv 3370 . . . . . . . . . . 11  |-  ( B  X.  C )  C_  _V
70 fnssres 5560 . . . . . . . . . . 11  |-  ( ( 1st  Fn  _V  /\  ( B  X.  C
)  C_  _V )  ->  ( 1st  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
7168, 69, 70mp2an 655 . . . . . . . . . 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 5599 . . . . . . . . . 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 5555 . . . . . . . . 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 1185 . . . . . . . 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 5802 . . . . . . . . . 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 459 . . . . . . . . 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 454 . . . . . . . . . 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 5734 . . . . . . . . 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 5870 . . . . . . . . . . . . . 14  |-  ( ( F : A --> B  /\  z  e.  A )  ->  ( F `  z
)  e.  B )
82 ffvelrn 5870 . . . . . . . . . . . . . 14  |-  ( ( G : A --> C  /\  z  e.  A )  ->  ( G `  z
)  e.  C )
83 opelxpi 4912 . . . . . . . . . . . . . 14  |-  ( ( ( F `  z
)  e.  B  /\  ( G `  z )  e.  C )  ->  <. ( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
) )
8481, 82, 83syl2an 465 . . . . . . . . . . . . 13  |-  ( ( ( F : A --> B  /\  z  e.  A
)  /\  ( G : A --> C  /\  z  e.  A ) )  ->  <. ( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
) )
8584anandirs 806 . . . . . . . . . . . 12  |-  ( ( ( F : A --> B  /\  G : A --> C )  /\  z  e.  A )  ->  <. ( F `  z ) ,  ( G `  z ) >.  e.  ( B  X.  C ) )
86853adantl1 1114 . . . . . . . . . . 11  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  <. ( F `
 z ) ,  ( G `  z
) >.  e.  ( B  X.  C ) )
87 fvres 5747 . . . . . . . . . . 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 5744 . . . . . . . . . . 11  |-  ( F `
 z )  e. 
_V
90 fvex 5744 . . . . . . . . . . 11  |-  ( G `
 z )  e. 
_V
9189, 90op1st 6357 . . . . . . . . . 10  |-  ( 1st `  <. ( F `  z ) ,  ( G `  z )
>. )  =  ( F `  z )
9288, 91syl6eq 2486 . . . . . . . . 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 2475 . . . . . . . 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 5832 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  =  ( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) )
9527coeq1i 5034 . . . . . . 7  |-  ( P  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  =  ( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
9694, 95syl6eqr 2488 . . . . . 6  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  =  ( P  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
97 ffn 5593 . . . . . . . . 9  |-  ( G : A --> C  ->  G  Fn  A )
98973ad2ant3 981 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  Fn  A
)
99 fo2nd 6369 . . . . . . . . . . . 12  |-  2nd : _V -onto-> _V
100 fofn 5657 . . . . . . . . . . . 12  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
10199, 100ax-mp 8 . . . . . . . . . . 11  |-  2nd  Fn  _V
102 fnssres 5560 . . . . . . . . . . 11  |-  ( ( 2nd  Fn  _V  /\  ( B  X.  C
)  C_  _V )  ->  ( 2nd  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
103101, 69, 102mp2an 655 . . . . . . . . . 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 5555 . . . . . . . . 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 1185 . . . . . . . 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 5802 . . . . . . . . . 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 459 . . . . . . . . 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 5734 . . . . . . . . 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 5747 . . . . . . . . . . 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 6358 . . . . . . . . . 10  |-  ( 2nd `  <. ( F `  z ) ,  ( G `  z )
>. )  =  ( G `  z )
113111, 112syl6eq 2486 . . . . . . . . 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 2475 . . . . . . . 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 5832 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) )
11642coeq1i 5034 . . . . . . 7  |-  ( Q  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
117115, 116syl6eqr 2488 . . . . . 6  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  =  ( Q  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
11817, 96, 1173jca 1135 . . . . 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 5578 . . . . . 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 5033 . . . . . . 7  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( P  o.  h )  =  ( P  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) )
121120eqeq2d 2449 . . . . . 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 5033 . . . . . . 7  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( Q  o.  h )  =  ( Q  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) )
123122eqeq2d 2449 . . . . . 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 1255 . . . . 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 215 . . . 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 185 . . 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 2291 . 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 225 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E!weu 2283   A.wral 2707   _Vcvv 2958    C_ wss 3322   <.cop 3819    e. cmpt 4268    X. cxp 4878   ran crn 4881    |` cres 4882    o. ccom 4884    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   ` cfv 5456   1stc1st 6349   2ndc2nd 6350
This theorem is referenced by:  uptx  17659  txcn  17660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1st 6351  df-2nd 6352
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