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Theorem upxp 20636
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 6150 . . . 4  |-  ( A  e.  D  ->  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  e.  _V )
2 eueq 3242 . . . 4  |-  ( ( x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
)  e.  _V  <->  E! h  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
)
31, 2sylib 199 . . 3  |-  ( A  e.  D  ->  E! h  h  =  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) )
433ad2ant1 1026 . 2  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  E! h  h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
5 ffn 5746 . . . . . . . 8  |-  ( h : A --> ( B  X.  C )  ->  h  Fn  A )
653ad2ant1 1026 . . . . . . 7  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  ->  h  Fn  A )
76adantl 467 . . . . . 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 6035 . . . . . . . . . . . . 13  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
9 ffvelrn 6035 . . . . . . . . . . . . 13  |-  ( ( G : A --> C  /\  x  e.  A )  ->  ( G `  x
)  e.  C )
10 opelxpi 4885 . . . . . . . . . . . . 13  |-  ( ( ( F `  x
)  e.  B  /\  ( G `  x )  e.  C )  ->  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
118, 9, 10syl2an 479 . . . . . . . . . . . 12  |-  ( ( ( F : A --> B  /\  x  e.  A
)  /\  ( G : A --> C  /\  x  e.  A ) )  ->  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
1211anandirs 838 . . . . . . . . . . 11  |-  ( ( ( F : A --> B  /\  G : A --> C )  /\  x  e.  A )  ->  <. ( F `  x ) ,  ( G `  x ) >.  e.  ( B  X.  C ) )
1312ralrimiva 2836 . . . . . . . . . 10  |-  ( ( F : A --> B  /\  G : A --> C )  ->  A. x  e.  A  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
14133adant1 1023 . . . . . . . . 9  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  A. x  e.  A  <. ( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )
15 eqid 2422 . . . . . . . . . 10  |-  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )
1615fmpt 6058 . . . . . . . . 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 199 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) : A --> ( B  X.  C ) )
18 ffn 5746 . . . . . . . 8  |-  ( ( x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) : A --> ( B  X.  C )  -> 
( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
)  Fn  A )
1917, 18syl 17 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  Fn  A
)
2019adantr 466 . . . . . 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 4960 . . . . . . . . . . 11  |-  ( B  X.  C )  C_  ( _V  X.  _V )
22 ffvelrn 6035 . . . . . . . . . . 11  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( h `  z
)  e.  ( B  X.  C ) )
2321, 22sseldi 3462 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( h `  z
)  e.  ( _V 
X.  _V ) )
24233ad2antl1 1167 . . . . . . . . 9  |-  ( ( ( h : A --> ( B  X.  C
)  /\  F  =  ( P  o.  h
)  /\  G  =  ( Q  o.  h
) )  /\  z  e.  A )  ->  (
h `  z )  e.  ( _V  X.  _V ) )
2524adantll 718 . . . . . . . 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 5880 . . . . . . . . . . . 12  |-  ( F  =  ( P  o.  h )  ->  ( F `  z )  =  ( ( P  o.  h ) `  z ) )
27 upxp.1 . . . . . . . . . . . . . 14  |-  P  =  ( 1st  |`  ( B  X.  C ) )
2827coeq1i 5013 . . . . . . . . . . . . 13  |-  ( P  o.  h )  =  ( ( 1st  |`  ( B  X.  C ) )  o.  h )
2928fveq1i 5882 . . . . . . . . . . . 12  |-  ( ( P  o.  h ) `
 z )  =  ( ( ( 1st  |`  ( B  X.  C
) )  o.  h
) `  z )
3026, 29syl6eq 2479 . . . . . . . . . . 11  |-  ( F  =  ( P  o.  h )  ->  ( F `  z )  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  h ) `  z
) )
31303ad2ant2 1027 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  -> 
( F `  z
)  =  ( ( ( 1st  |`  ( B  X.  C ) )  o.  h ) `  z ) )
3231ad2antlr 731 . . . . . . . . 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 1011 . . . . . . . . . 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 5958 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 1st  |`  ( B  X.  C
) )  o.  h
) `  z )  =  ( ( 1st  |`  ( B  X.  C
) ) `  (
h `  z )
) )
3533, 34sylan 473 . . . . . . . . 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 1167 . . . . . . . . . . 11  |-  ( ( ( h : A --> ( B  X.  C
)  /\  F  =  ( P  o.  h
)  /\  G  =  ( Q  o.  h
) )  /\  z  e.  A )  ->  (
h `  z )  e.  ( B  X.  C
) )
3736adantll 718 . . . . . . . . . 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 5895 . . . . . . . . . 10  |-  ( ( h `  z )  e.  ( B  X.  C )  ->  (
( 1st  |`  ( B  X.  C ) ) `
 ( h `  z ) )  =  ( 1st `  (
h `  z )
) )
3937, 38syl 17 . . . . . . . . 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 2468 . . . . . . . 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 5880 . . . . . . . . . . . 12  |-  ( G  =  ( Q  o.  h )  ->  ( G `  z )  =  ( ( Q  o.  h ) `  z ) )
42 upxp.2 . . . . . . . . . . . . . 14  |-  Q  =  ( 2nd  |`  ( B  X.  C ) )
4342coeq1i 5013 . . . . . . . . . . . . 13  |-  ( Q  o.  h )  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  h )
4443fveq1i 5882 . . . . . . . . . . . 12  |-  ( ( Q  o.  h ) `
 z )  =  ( ( ( 2nd  |`  ( B  X.  C
) )  o.  h
) `  z )
4541, 44syl6eq 2479 . . . . . . . . . . 11  |-  ( G  =  ( Q  o.  h )  ->  ( G `  z )  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z
) )
46453ad2ant3 1028 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) )  -> 
( G `  z
)  =  ( ( ( 2nd  |`  ( B  X.  C ) )  o.  h ) `  z ) )
4746ad2antlr 731 . . . . . . . . 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 5958 . . . . . . . . . 10  |-  ( ( h : A --> ( B  X.  C )  /\  z  e.  A )  ->  ( ( ( 2nd  |`  ( B  X.  C
) )  o.  h
) `  z )  =  ( ( 2nd  |`  ( B  X.  C
) ) `  (
h `  z )
) )
4933, 48sylan 473 . . . . . . . . 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 5895 . . . . . . . . . 10  |-  ( ( h `  z )  e.  ( B  X.  C )  ->  (
( 2nd  |`  ( B  X.  C ) ) `
 ( h `  z ) )  =  ( 2nd `  (
h `  z )
) )
5137, 50syl 17 . . . . . . . . 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 2468 . . . . . . . 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 6841 . . . . . . . 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 1262 . . . . . . 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 5881 . . . . . . . . . 10  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
56 fveq2 5881 . . . . . . . . . 10  |-  ( x  =  z  ->  ( G `  x )  =  ( G `  z ) )
5755, 56opeq12d 4195 . . . . . . . . 9  |-  ( x  =  z  ->  <. ( F `  x ) ,  ( G `  x ) >.  =  <. ( F `  z ) ,  ( G `  z ) >. )
58 opex 4685 . . . . . . . . 9  |-  <. ( F `  z ) ,  ( G `  z ) >.  e.  _V
5957, 15, 58fvmpt 5964 . . . . . . . 8  |-  ( z  e.  A  ->  (
( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
) `  z )  =  <. ( F `  z ) ,  ( G `  z )
>. )
6059adantl 467 . . . . . . 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 2466 . . . . . 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 5994 . . . . 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 435 . . . 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 5746 . . . . . . . . 9  |-  ( F : A --> B  ->  F  Fn  A )
65643ad2ant2 1027 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  Fn  A
)
66 fo1st 6827 . . . . . . . . . . . 12  |-  1st : _V -onto-> _V
67 fofn 5812 . . . . . . . . . . . 12  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
6866, 67ax-mp 5 . . . . . . . . . . 11  |-  1st  Fn  _V
69 ssv 3484 . . . . . . . . . . 11  |-  ( B  X.  C )  C_  _V
70 fnssres 5707 . . . . . . . . . . 11  |-  ( ( 1st  Fn  _V  /\  ( B  X.  C
)  C_  _V )  ->  ( 1st  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
7168, 69, 70mp2an 676 . . . . . . . . . 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 5752 . . . . . . . . . 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 17 . . . . . . . . 9  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  ran  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )  C_  ( B  X.  C
) )
75 fnco 5702 . . . . . . . . 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 1264 . . . . . . . 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 5958 . . . . . . . . . 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 473 . . . . . . . . 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 467 . . . . . . . . . 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 5885 . . . . . . . . 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 6035 . . . . . . . . . . . . . 14  |-  ( ( F : A --> B  /\  z  e.  A )  ->  ( F `  z
)  e.  B )
82 ffvelrn 6035 . . . . . . . . . . . . . 14  |-  ( ( G : A --> C  /\  z  e.  A )  ->  ( G `  z
)  e.  C )
83 opelxpi 4885 . . . . . . . . . . . . . 14  |-  ( ( ( F `  z
)  e.  B  /\  ( G `  z )  e.  C )  ->  <. ( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
) )
8481, 82, 83syl2an 479 . . . . . . . . . . . . 13  |-  ( ( ( F : A --> B  /\  z  e.  A
)  /\  ( G : A --> C  /\  z  e.  A ) )  ->  <. ( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
) )
8584anandirs 838 . . . . . . . . . . . 12  |-  ( ( ( F : A --> B  /\  G : A --> C )  /\  z  e.  A )  ->  <. ( F `  z ) ,  ( G `  z ) >.  e.  ( B  X.  C ) )
86853adantl1 1161 . . . . . . . . . . 11  |-  ( ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  /\  z  e.  A
)  ->  <. ( F `
 z ) ,  ( G `  z
) >.  e.  ( B  X.  C ) )
87 fvres 5895 . . . . . . . . . . 11  |-  ( <.
( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
)  ->  ( ( 1st  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( 1st `  <. ( F `  z ) ,  ( G `  z )
>. ) )
8886, 87syl 17 . . . . . . . . . 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 5891 . . . . . . . . . . 11  |-  ( F `
 z )  e. 
_V
90 fvex 5891 . . . . . . . . . . 11  |-  ( G `
 z )  e. 
_V
9189, 90op1st 6815 . . . . . . . . . 10  |-  ( 1st `  <. ( F `  z ) ,  ( G `  z )
>. )  =  ( F `  z )
9288, 91syl6eq 2479 . . . . . . . . 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 2468 . . . . . . . 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 5994 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  =  ( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) )
9527coeq1i 5013 . . . . . . 7  |-  ( P  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  =  ( ( 1st  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
9694, 95syl6eqr 2481 . . . . . 6  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  F  =  ( P  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
97 ffn 5746 . . . . . . . . 9  |-  ( G : A --> C  ->  G  Fn  A )
98973ad2ant3 1028 . . . . . . . 8  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  Fn  A
)
99 fo2nd 6828 . . . . . . . . . . . 12  |-  2nd : _V -onto-> _V
100 fofn 5812 . . . . . . . . . . . 12  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
10199, 100ax-mp 5 . . . . . . . . . . 11  |-  2nd  Fn  _V
102 fnssres 5707 . . . . . . . . . . 11  |-  ( ( 2nd  Fn  _V  /\  ( B  X.  C
)  C_  _V )  ->  ( 2nd  |`  ( B  X.  C ) )  Fn  ( B  X.  C ) )
103101, 69, 102mp2an 676 . . . . . . . . . 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 5702 . . . . . . . . 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 1264 . . . . . . . 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 5958 . . . . . . . . . 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 473 . . . . . . . . 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 5885 . . . . . . . . 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 5895 . . . . . . . . . . 11  |-  ( <.
( F `  z
) ,  ( G `
 z ) >.  e.  ( B  X.  C
)  ->  ( ( 2nd  |`  ( B  X.  C ) ) `  <. ( F `  z
) ,  ( G `
 z ) >.
)  =  ( 2nd `  <. ( F `  z ) ,  ( G `  z )
>. ) )
11186, 110syl 17 . . . . . . . . . 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 6816 . . . . . . . . . 10  |-  ( 2nd `  <. ( F `  z ) ,  ( G `  z )
>. )  =  ( G `  z )
113111, 112syl6eq 2479 . . . . . . . . 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 2468 . . . . . . . 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 5994 . . . . . . 7  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) ) )
11642coeq1i 5013 . . . . . . 7  |-  ( Q  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )  =  ( ( 2nd  |`  ( B  X.  C ) )  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. ) )
117115, 116syl6eqr 2481 . . . . . 6  |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  G  =  ( Q  o.  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )
) )
11817, 96, 1173jca 1185 . . . . 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 5728 . . . . . 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 5012 . . . . . . 7  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( P  o.  h )  =  ( P  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) )
121120eqeq2d 2436 . . . . . 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 5012 . . . . . . 7  |-  ( h  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x )
>. )  ->  ( Q  o.  h )  =  ( Q  o.  (
x  e.  A  |->  <.
( F `  x
) ,  ( G `
 x ) >.
) ) )
123122eqeq2d 2436 . . . . . 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 1335 . . . . 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 225 . . . 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 193 . . 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 2289 . 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 235 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 370    /\ w3a 982    = wceq 1437    e. wcel 1872   E!weu 2269   A.wral 2771   _Vcvv 3080    C_ wss 3436   <.cop 4004    |-> cmpt 4482    X. cxp 4851   ran crn 4854    |` cres 4855    o. ccom 4857    Fn wfn 5596   -->wf 5597   -onto->wfo 5599   ` cfv 5601   1stc1st 6805   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-1st 6807  df-2nd 6808
This theorem is referenced by:  uptx  20638  txcn  20639
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