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Theorem updjud 6933
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 302 . . . . 5  |-  ( ph  ->  ( A  e.  V  /\  B  e.  W
) )
4 djuex 6894 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  e.  _V )
5 mptexg 5611 . . . . 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 5223 . . . . . . 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 4664 . . . . . . . 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 2124 . . . . . . 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 4664 . . . . . . . 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 2124 . . . . . . 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 1273 . . . . . 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 2122 . . . . . . . 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 229 . . . . . . 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 2412 . . . . . 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 462 . . . . 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 273 . . . 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 2115 . . . . . 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 6930 . . . . 5  |-  ( ph  ->  ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) : ( A B ) --> C )
2218, 19, 20updjudhcoinlf 6931 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inl  |`  A ) )  =  F )
2318, 19, 20updjudhcoinrg 6932 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )  o.  (inr  |`  B ) )  =  G )
24 simpr 109 . . . . . . 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 2125 . . . . . . . . . . . . . . . . . . . . . 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 5411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( z  e.  A  ->  (
(inl  |`  A ) `  z )  =  (inl
`  z ) )
2726eqcomd 2121 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( z  e.  A  ->  (inl `  z )  =  ( (inl  |`  A ) `  z ) )
2827eqeq2d 2127 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  e.  A  ->  (
y  =  (inl `  z )  <->  y  =  ( (inl  |`  A ) `
 z ) ) )
2928adantl 273 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5386 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  (inl  |`  A ) : A -1-1-> ( A B )
33 f1fn 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( (inl  |`  A )  Fn  A  /\  z  e.  A )  ->  (
( k  o.  (inl  |`  A ) ) `  z )  =  ( k `  ( (inl  |`  A ) `  z
) ) )
3834, 37sylan 279 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2156 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5387 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5387 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( y  =  ( (inl  |`  A ) `
 z )  -> 
( k `  y
)  =  ( k `
 ( (inl  |`  A ) `
 z ) ) )
4240, 41eqeq12d 2130 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 156 . . . . . . . . . . . . . . . . . . . . . . . . . 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 149 . . . . . . . . . . . . . . . . . . . . . . . . 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 358 . . . . . . . . . . . . . . . . . . . . . . . 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 114 . . . . . . . . . . . . . . . . . . . . . . 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 2118 . . . . . . . . . . . . . . . . . . . . . 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 163 . . . . . . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . . . 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 2519 . . . . . . . . . . . . 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 2125 . . . . . . . . . . . . . . . . . . . . . 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 5411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( z  e.  B  ->  (
(inr  |`  B ) `  z )  =  (inr
`  z ) )
5958eqcomd 2121 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( z  e.  B  ->  (inr `  z )  =  ( (inr  |`  B ) `  z ) )
6059eqeq2d 2127 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  e.  B  ->  (
y  =  (inr `  z )  <->  y  =  ( (inr  |`  B ) `
 z ) ) )
6160adantl 273 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5386 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  (inr  |`  B ) : B -1-1-> ( A B )
65 f1fn 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( (inr  |`  B )  Fn  B  /\  z  e.  B )  ->  (
( k  o.  (inr  |`  B ) ) `  z )  =  ( k `  ( (inr  |`  B ) `  z
) ) )
7066, 69sylan 279 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2156 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5387 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5387 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( y  =  ( (inr  |`  B ) `
 z )  -> 
( k `  y
)  =  ( k `
 ( (inr  |`  B ) `
 z ) ) )
7472, 73eqeq12d 2130 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 156 . . . . . . . . . . . . . . . . . . . . . . . . . 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 149 . . . . . . . . . . . . . . . . . . . . . . . . 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 358 . . . . . . . . . . . . . . . . . . . . . . . 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 114 . . . . . . . . . . . . . . . . . . . . . . 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 2118 . . . . . . . . . . . . . . . . . . . . . 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 163 . . . . . . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . . . . . . 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 987 . . . . . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . . . . . . 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 987 . . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . . . 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 2519 . . . . . . . . . . . . 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 688 . . . . . . . . . . . 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 6920 . . . . . . . . . . . . 13  |-  ( y  e.  ( A B )  <-> 
( E. z  e.  A  y  =  (inl
`  z )  \/ 
E. z  e.  B  y  =  (inr `  z
) ) )
9190biimpi 119 . . . . . . . . . . . 12  |-  ( y  e.  ( A B )  ->  ( E. z  e.  A  y  =  (inl `  z )  \/ 
E. z  e.  B  y  =  (inr `  z
) ) )
9289, 91syl11 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 ) ) )
9392ralrimiv 2479 . . . . . . . . . 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 ) )
94 ffn 5240 . . . . . . . . . . . . 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 ) )
95943ad2ant1 985 . . . . . . . . . . . 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 ) )
9695adantl 273 . . . . . . . . . . 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 ) )
97 ffn 5240 . . . . . . . . . . . 12  |-  ( k : ( A B ) --> C  ->  k  Fn  ( A B ) )
98973ad2ant1 985 . . . . . . . . . . 11  |-  ( ( k : ( A B ) --> C  /\  ( k  o.  (inl  |`  A ) )  =  F  /\  ( k  o.  (inr  |`  B ) )  =  G )  ->  k  Fn  ( A B ) )
99 eqfnfv 5484 . . . . . . . . . . 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 ) ) )
10096, 98, 99syl2an 285 . . . . . . . . . 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 ) ) )
10193, 100mpbird 166 . . . . . . . . 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 )
102101ex 114 . . . . . . . 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 ) )
103102ralrimivw 2481 . . . . . . 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 ) )
10424, 103jca 302 . . . . . 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 ) ) )
105104ex 114 . . . . 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 ) ) ) )
10621, 22, 23, 105mp3and 1301 . . . 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 ) ) )
1076, 17, 106rspcedvd 2767 . . 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 ) ) )
108 feq1 5223 . . . . 5  |-  ( h  =  k  ->  (
h : ( A B ) --> C  <->  k :
( A B ) --> C ) )
109 coeq1 4664 . . . . . 6  |-  ( h  =  k  ->  (
h  o.  (inl  |`  A ) )  =  ( k  o.  (inl  |`  A ) ) )
110109eqeq1d 2124 . . . . 5  |-  ( h  =  k  ->  (
( h  o.  (inl  |`  A ) )  =  F  <->  ( k  o.  (inl  |`  A ) )  =  F ) )
111 coeq1 4664 . . . . . 6  |-  ( h  =  k  ->  (
h  o.  (inr  |`  B ) )  =  ( k  o.  (inr  |`  B ) ) )
112111eqeq1d 2124 . . . . 5  |-  ( h  =  k  ->  (
( h  o.  (inr  |`  B ) )  =  G  <->  ( k  o.  (inr  |`  B ) )  =  G ) )
113108, 110, 1123anbi123d 1273 . . . 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 ) ) )
114113reu8 2851 . . 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 ) ) )
115107, 114sylibr 133 . 2  |-  ( ph  ->  E! h  e.  _V  ( h : ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G ) )
116 reuv 2677 . 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 ) )
117115, 116sylib 121 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 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   E!weu 1975   A.wral 2391   E.wrex 2392   E!wreu 2393   _Vcvv 2658   (/)c0 3331   ifcif 3442    |-> cmpt 3957    |` cres 4509    o. ccom 4511    Fn wfn 5086   -->wf 5087   -1-1->wf1 5088   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   ⊔ cdju 6888  inlcinl 6896  inrcinr 6897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899
This theorem is referenced by: (None)
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