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Theorem updjud 6752
Description: Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
Hypotheses
Ref Expression
updjud.f  |-  ( ph  ->  F : A --> C )
updjud.g  |-  ( ph  ->  G : B --> C )
updjud.a  |-  ( ph  ->  A  e.  V )
updjud.b  |-  ( ph  ->  B  e.  W )
Assertion
Ref Expression
updjud  |-  ( ph  ->  E! h ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G ) )
Distinct variable groups:    A, h    B, h    C, h    h, F   
h, G    ph, h
Allowed substitution hints:    V( h)    W( h)

Proof of Theorem updjud
Dummy variables  k  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 updjud.a . . . . . 6  |-  ( ph  ->  A  e.  V )
2 updjud.b . . . . . 6  |-  ( ph  ->  B  e.  W )
31, 2jca 300 . . . . 5  |-  ( ph  ->  ( A  e.  V  /\  B  e.  W
) )
4 djuex 6715 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  e.  _V )
5 mptexg 5504 . . . . 5  |-  ( ( A B )  e.  _V  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  e.  _V )
63, 4, 53syl 17 . . . 4  |-  ( ph  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  e.  _V )
7 feq1 5131 . . . . . . 7  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( h : ( A B ) --> C  <->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C ) )
8 coeq1 4581 . . . . . . . 8  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( h  o.  (inl  |`  A ) )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) ) )
98eqeq1d 2096 . . . . . . 7  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( ( h  o.  (inl  |`  A ) )  =  F  <->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F ) )
10 coeq1 4581 . . . . . . . 8  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( h  o.  (inr  |`  B ) )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) ) )
1110eqeq1d 2096 . . . . . . 7  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( ( h  o.  (inr  |`  B ) )  =  G  <->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )
127, 9, 113anbi123d 1248 . . . . . 6  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  (
h  o.  (inr  |`  B ) )  =  G )  <-> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) ) )
13 eqeq1 2094 . . . . . . . 8  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( h  =  k  <-> 
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) )
1413imbi2d 228 . . . . . . 7  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  h  =  k )  <->  ( (
k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) ) )
1514ralbidv 2380 . . . . . 6  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( A. k  e. 
_V  ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  h  =  k )  <->  A. k  e.  _V  ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) ) )
1612, 15anbi12d 457 . . . . 5  |-  ( h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  -> 
( ( ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G )  /\  A. k  e.  _V  (
( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  h  =  k ) )  <->  ( (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )  /\  A. k  e.  _V  (
( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) ) ) )
1716adantl 271 . . . 4  |-  ( (
ph  /\  h  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) )  -> 
( ( ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G )  /\  A. k  e.  _V  (
( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  h  =  k ) )  <->  ( (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )  /\  A. k  e.  _V  (
( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) ) ) )
18 updjud.f . . . . . 6  |-  ( ph  ->  F : A --> C )
19 updjud.g . . . . . 6  |-  ( ph  ->  G : B --> C )
20 eqid 2088 . . . . . 6  |-  ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )
2118, 19, 20updjudhf 6749 . . . . 5  |-  ( ph  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C )
2218, 19, 20updjudhcoinlf 6750 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F )
2318, 19, 20updjudhcoinrg 6751 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )
24 simpr 108 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )
25 eqeq2 2097 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  ->  ( (
k  o.  (inl  |`  A ) )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  <->  ( k  o.  (inl  |`  A ) )  =  F ) )
26 fvres 5313 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( z  e.  A  ->  (
(inl  |`  A ) `  z )  =  (inl
`  z ) )
2726eqcomd 2093 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( z  e.  A  ->  (inl `  z )  =  ( (inl  |`  A ) `  z ) )
2827eqeq2d 2099 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  e.  A  ->  (
y  =  (inl `  z )  <->  y  =  ( (inl  |`  A ) `
 z ) ) )
2928adantl 271 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  /\  ph )  /\  z  e.  A )  ->  ( y  =  (inl
`  z )  <->  y  =  ( (inl  |`  A ) `
 z ) ) )
30 fveq1 5288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  -> 
( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) ) `
 z )  =  ( ( k  o.  (inl  |`  A ) ) `
 z ) )
3130ad2antrr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  /\  ph )  /\  z  e.  A )  ->  ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) ) `
 z )  =  ( ( k  o.  (inl  |`  A ) ) `
 z ) )
32 inlresf1 6732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  (inl  |`  A ) : A -1-1-> ( A B )
33 f1fn 5202 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (inl  |`  A ) : A -1-1-> ( A B )  ->  (inl  |`  A )  Fn  A
)
3432, 33mp1i 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  /\  ph )  -> 
(inl  |`  A )  Fn  A )
35 fvco2 5357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( (inl  |`  A )  Fn  A  /\  z  e.  A )  ->  (
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) ) `
 z )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  ( (inl  |`  A ) `
 z ) ) )
3634, 35sylan 277 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  /\  ph )  /\  z  e.  A )  ->  ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) ) `
 z )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  ( (inl  |`  A ) `
 z ) ) )
37 fvco2 5357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( (inl  |`  A )  Fn  A  /\  z  e.  A )  ->  (
( k  o.  (inl  |`  A ) ) `  z )  =  ( k `  ( (inl  |`  A ) `  z
) ) )
3834, 37sylan 277 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  /\  ph )  /\  z  e.  A )  ->  ( ( k  o.  (inl  |`  A ) ) `
 z )  =  ( k `  (
(inl  |`  A ) `  z ) ) )
3931, 36, 383eqtr3d 2128 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  /\  ph )  /\  z  e.  A )  ->  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  ( (inl  |`  A ) `
 z ) )  =  ( k `  ( (inl  |`  A ) `
 z ) ) )
40 fveq2 5289 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( y  =  ( (inl  |`  A ) `
 z )  -> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  (
(inl  |`  A ) `  z ) ) )
41 fveq2 5289 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( y  =  ( (inl  |`  A ) `
 z )  -> 
( k `  y
)  =  ( k `
 ( (inl  |`  A ) `
 z ) ) )
4240, 41eqeq12d 2102 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( y  =  ( (inl  |`  A ) `
 z )  -> 
( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y )  <-> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  ( (inl  |`  A ) `
 z ) )  =  ( k `  ( (inl  |`  A ) `
 z ) ) ) )
4339, 42syl5ibrcom 155 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  /\  ph )  /\  z  e.  A )  ->  ( y  =  ( (inl  |`  A ) `  z )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
4429, 43sylbid 148 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  /\  ph )  /\  z  e.  A )  ->  ( y  =  (inl
`  z )  -> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y ) ) )
4544expimpd 355 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  /\  ph )  -> 
( ( z  e.  A  /\  y  =  (inl `  z )
)  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
4645ex 113 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) )  -> 
( ph  ->  ( ( z  e.  A  /\  y  =  (inl `  z
) )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) )
4746eqcoms 2091 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( k  o.  (inl  |`  A ) )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  -> 
( ph  ->  ( ( z  e.  A  /\  y  =  (inl `  z
) )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) )
4825, 47syl6bir 162 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  ->  ( (
k  o.  (inl  |`  A ) )  =  F  -> 
( ph  ->  ( ( z  e.  A  /\  y  =  (inl `  z
) )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) ) )
4948com23 77 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  ->  ( ph  ->  ( ( k  o.  (inl  |`  A ) )  =  F  ->  (
( z  e.  A  /\  y  =  (inl `  z ) )  -> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y ) ) ) ) )
50493ad2ant2 965 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )  ->  ( ph  ->  ( ( k  o.  (inl  |`  A ) )  =  F  -> 
( ( z  e.  A  /\  y  =  (inl `  z )
)  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) ) )
5150impcom 123 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( ( k  o.  (inl  |`  A ) )  =  F  ->  (
( z  e.  A  /\  y  =  (inl `  z ) )  -> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y ) ) ) )
5251com12 30 . . . . . . . . . . . . . . . . 17  |-  ( ( k  o.  (inl  |`  A ) )  =  F  -> 
( ( ph  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( ( z  e.  A  /\  y  =  (inl `  z )
)  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) )
53523ad2ant2 965 . . . . . . . . . . . . . . . 16  |-  ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  ( ( ph  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( ( z  e.  A  /\  y  =  (inl `  z )
)  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) )
5453impcom 123 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( (
z  e.  A  /\  y  =  (inl `  z
) )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
5554com12 30 . . . . . . . . . . . . . 14  |-  ( ( z  e.  A  /\  y  =  (inl `  z
) )  ->  (
( ( ph  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
5655rexlimiva 2484 . . . . . . . . . . . . 13  |-  ( E. z  e.  A  y  =  (inl `  z
)  ->  ( (
( ph  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
57 eqeq2 2097 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G  ->  ( (
k  o.  (inr  |`  B ) )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  <->  ( k  o.  (inr  |`  B ) )  =  G ) )
58 fvres 5313 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( z  e.  B  ->  (
(inr  |`  B ) `  z )  =  (inr
`  z ) )
5958eqcomd 2093 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( z  e.  B  ->  (inr `  z )  =  ( (inr  |`  B ) `  z ) )
6059eqeq2d 2099 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  e.  B  ->  (
y  =  (inr `  z )  <->  y  =  ( (inr  |`  B ) `
 z ) ) )
6160adantl 271 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  /\  ph )  /\  z  e.  B )  ->  ( y  =  (inr
`  z )  <->  y  =  ( (inr  |`  B ) `
 z ) ) )
62 fveq1 5288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  -> 
( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) ) `
 z )  =  ( ( k  o.  (inr  |`  B ) ) `
 z ) )
6362ad2antrr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  /\  ph )  /\  z  e.  B )  ->  ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) ) `
 z )  =  ( ( k  o.  (inr  |`  B ) ) `
 z ) )
64 inrresf1 6733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  (inr  |`  B ) : B -1-1-> ( A B )
65 f1fn 5202 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (inr  |`  B ) : B -1-1-> ( A B )  ->  (inr  |`  B )  Fn  B
)
6664, 65mp1i 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  /\  ph )  -> 
(inr  |`  B )  Fn  B )
67 fvco2 5357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( (inr  |`  B )  Fn  B  /\  z  e.  B )  ->  (
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) ) `
 z )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  ( (inr  |`  B ) `
 z ) ) )
6866, 67sylan 277 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  /\  ph )  /\  z  e.  B )  ->  ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) ) `
 z )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  ( (inr  |`  B ) `
 z ) ) )
69 fvco2 5357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( (inr  |`  B )  Fn  B  /\  z  e.  B )  ->  (
( k  o.  (inr  |`  B ) ) `  z )  =  ( k `  ( (inr  |`  B ) `  z
) ) )
7066, 69sylan 277 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  /\  ph )  /\  z  e.  B )  ->  ( ( k  o.  (inr  |`  B ) ) `
 z )  =  ( k `  (
(inr  |`  B ) `  z ) ) )
7163, 68, 703eqtr3d 2128 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  /\  ph )  /\  z  e.  B )  ->  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  ( (inr  |`  B ) `
 z ) )  =  ( k `  ( (inr  |`  B ) `
 z ) ) )
72 fveq2 5289 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( y  =  ( (inr  |`  B ) `
 z )  -> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  (
(inr  |`  B ) `  z ) ) )
73 fveq2 5289 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( y  =  ( (inr  |`  B ) `
 z )  -> 
( k `  y
)  =  ( k `
 ( (inr  |`  B ) `
 z ) ) )
7472, 73eqeq12d 2102 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( y  =  ( (inr  |`  B ) `
 z )  -> 
( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y )  <-> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  ( (inr  |`  B ) `
 z ) )  =  ( k `  ( (inr  |`  B ) `
 z ) ) ) )
7571, 74syl5ibrcom 155 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  /\  ph )  /\  z  e.  B )  ->  ( y  =  ( (inr  |`  B ) `  z )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
7661, 75sylbid 148 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  /\  ph )  /\  z  e.  B )  ->  ( y  =  (inr
`  z )  -> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y ) ) )
7776expimpd 355 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  /\  ph )  -> 
( ( z  e.  B  /\  y  =  (inr `  z )
)  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
7877ex 113 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) )  -> 
( ph  ->  ( ( z  e.  B  /\  y  =  (inr `  z
) )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) )
7978eqcoms 2091 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( k  o.  (inr  |`  B ) )  =  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  -> 
( ph  ->  ( ( z  e.  B  /\  y  =  (inr `  z
) )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) )
8057, 79syl6bir 162 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G  ->  ( (
k  o.  (inr  |`  B ) )  =  G  -> 
( ph  ->  ( ( z  e.  B  /\  y  =  (inr `  z
) )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) ) )
8180com23 77 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G  ->  ( ph  ->  ( ( k  o.  (inr  |`  B ) )  =  G  ->  (
( z  e.  B  /\  y  =  (inr `  z ) )  -> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y ) ) ) ) )
82813ad2ant3 966 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )  ->  ( ph  ->  ( ( k  o.  (inr  |`  B ) )  =  G  -> 
( ( z  e.  B  /\  y  =  (inr `  z )
)  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) ) )
8382impcom 123 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( ( k  o.  (inr  |`  B ) )  =  G  ->  (
( z  e.  B  /\  y  =  (inr `  z ) )  -> 
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y ) ) ) )
8483com12 30 . . . . . . . . . . . . . . . . 17  |-  ( ( k  o.  (inr  |`  B ) )  =  G  -> 
( ( ph  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( ( z  e.  B  /\  y  =  (inr `  z )
)  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) )
85843ad2ant3 966 . . . . . . . . . . . . . . . 16  |-  ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  ( ( ph  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( ( z  e.  B  /\  y  =  (inr `  z )
)  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) ) )
8685impcom 123 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( (
z  e.  B  /\  y  =  (inr `  z
) )  ->  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
8786com12 30 . . . . . . . . . . . . . 14  |-  ( ( z  e.  B  /\  y  =  (inr `  z
) )  ->  (
( ( ph  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
8887rexlimiva 2484 . . . . . . . . . . . . 13  |-  ( E. z  e.  B  y  =  (inr `  z
)  ->  ( (
( ph  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
8956, 88jaoi 671 . . . . . . . . . . . 12  |-  ( ( E. z  e.  A  y  =  (inl `  z
)  \/  E. z  e.  B  y  =  (inr `  z ) )  ->  ( ( (
ph  /\  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
90 djur 6736 . . . . . . . . . . . 12  |-  ( y  e.  ( A B )  ->  ( E. z  e.  A  y  =  (inl `  z )  \/ 
E. z  e.  B  y  =  (inr `  z
) ) )
9189, 90syl11 31 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( y  e.  ( A B )  ->  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y ) ) )
9291ralrimiv 2445 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  A. y  e.  ( A B )
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y )  =  ( k `  y ) )
93 ffn 5147 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  -> 
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  Fn  ( A B ) )
94933ad2ant1 964 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )  ->  (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  Fn  ( A B ) )
9594adantl 271 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  Fn  ( A B ) )
96 ffn 5147 . . . . . . . . . . . 12  |-  ( k : ( A B ) --> C  ->  k  Fn  ( A B ) )
97963ad2ant1 964 . . . . . . . . . . 11  |-  ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  k  Fn  ( A B ) )
98 eqfnfv 5381 . . . . . . . . . . 11  |-  ( ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  Fn  ( A B )  /\  k  Fn  ( A B )
)  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k  <->  A. y  e.  ( A B ) ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
9995, 97, 98syl2an 283 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k  <->  A. y  e.  ( A B ) ( ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) `  y
)  =  ( k `
 y ) ) )
10092, 99mpbird 165 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  /\  ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) )  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k )
101100ex 113 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  (
k  o.  (inr  |`  B ) )  =  G )  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) )
102101ralrimivw 2447 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  ->  A. k  e.  _V  ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  (
k  o.  (inr  |`  B ) )  =  G )  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) )
10324, 102jca 300 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G ) )  -> 
( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )  /\  A. k  e.  _V  (
( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) ) )
104103ex 113 . . . . 5  |-  ( ph  ->  ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )  ->  (
( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )  /\  A. k  e.  _V  (
( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) ) ) )
10521, 22, 23, 104mp3and 1276 . . . 4  |-  ( ph  ->  ( ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C  /\  ( ( x  e.  ( A B ) 
|->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F  /\  (
( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )  /\  A. k  e.  _V  (
( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  =  k ) ) )
1066, 17, 105rspcedvd 2728 . . 3  |-  ( ph  ->  E. h  e.  _V  ( ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  (
h  o.  (inr  |`  B ) )  =  G )  /\  A. k  e. 
_V  ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  h  =  k ) ) )
107 feq1 5131 . . . . 5  |-  ( h  =  k  ->  (
h : ( A B ) --> C  <->  k :
( A B ) --> C ) )
108 coeq1 4581 . . . . . 6  |-  ( h  =  k  ->  (
h  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) ) )
109108eqeq1d 2096 . . . . 5  |-  ( h  =  k  ->  (
( h  o.  (inl  |`  A ) )  =  F  <->  ( k  o.  (inl  |`  A ) )  =  F ) )
110 coeq1 4581 . . . . . 6  |-  ( h  =  k  ->  (
h  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) ) )
111110eqeq1d 2096 . . . . 5  |-  ( h  =  k  ->  (
( h  o.  (inr  |`  B ) )  =  G  <->  ( k  o.  (inr  |`  B ) )  =  G ) )
112107, 109, 1113anbi123d 1248 . . . 4  |-  ( h  =  k  ->  (
( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G )  <-> 
( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G ) ) )
113112reu8 2809 . . 3  |-  ( E! h  e.  _V  (
h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G )  <->  E. h  e.  _V  ( ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  (
h  o.  (inr  |`  B ) )  =  G )  /\  A. k  e. 
_V  ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  h  =  k ) ) )
114106, 113sylibr 132 . 2  |-  ( ph  ->  E! h  e.  _V  ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G ) )
115 reuv 2638 . 2  |-  ( E! h  e.  _V  (
h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G )  <-> 
E! h ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G ) )
116114, 115sylib 120 1  |-  ( ph  ->  E! h ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   E!weu 1948   A.wral 2359   E.wrex 2360   E!wreu 2361   _Vcvv 2619   (/)c0 3284   ifcif 3389    |-> cmpt 3891    |` cres 4430    o. ccom 4432    Fn wfn 4997   -->wf 4998   -1-1->wf1 4999   ` cfv 5002   1stc1st 5891   2ndc2nd 5892   ⊔ cdju 6709  inlcinl 6716  inrcinr 6717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-1st 5893  df-2nd 5894  df-1o 6163  df-dju 6710  df-inl 6718  df-inr 6719
This theorem is referenced by: (None)
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