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Theorem natpropd 13866
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
fucpropd.1  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
fucpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
fucpropd.3  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
fucpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
fucpropd.a  |-  ( ph  ->  A  e.  Cat )
fucpropd.b  |-  ( ph  ->  B  e.  Cat )
fucpropd.c  |-  ( ph  ->  C  e.  Cat )
fucpropd.d  |-  ( ph  ->  D  e.  Cat )
Assertion
Ref Expression
natpropd  |-  ( ph  ->  ( A Nat  C )  =  ( B Nat  D
) )

Proof of Theorem natpropd
Dummy variables  a 
f  g  h  r  s  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucpropd.1 . . . 4  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
2 fucpropd.2 . . . 4  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
3 fucpropd.3 . . . 4  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
4 fucpropd.4 . . . 4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
5 fucpropd.a . . . 4  |-  ( ph  ->  A  e.  Cat )
6 fucpropd.b . . . 4  |-  ( ph  ->  B  e.  Cat )
7 fucpropd.c . . . 4  |-  ( ph  ->  C  e.  Cat )
8 fucpropd.d . . . 4  |-  ( ph  ->  D  e.  Cat )
91, 2, 3, 4, 5, 6, 7, 8funcpropd 13790 . . 3  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
109adantr 451 . . 3  |-  ( (
ph  /\  f  e.  ( A  Func  C ) )  ->  ( A  Func  C )  =  ( B  Func  D )
)
11 nfv 1609 . . . 4  |-  F/ r ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )
12 nfcsb1v 3126 . . . . 5  |-  F/_ r [_ ( 1st `  f
)  /  r ]_ [_ ( 1st `  g
)  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) }
1312a1i 10 . . . 4  |-  ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  ->  F/_ r [_ ( 1st `  f
)  /  r ]_ [_ ( 1st `  g
)  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
14 fvex 5555 . . . . 5  |-  ( 1st `  f )  e.  _V
1514a1i 10 . . . 4  |-  ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  ->  ( 1st `  f )  e. 
_V )
16 nfv 1609 . . . . . 6  |-  F/ s ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )
17 nfcsb1v 3126 . . . . . . 7  |-  F/_ s [_ ( 1st `  g
)  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) }
1817a1i 10 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  ->  F/_ s [_ ( 1st `  g )  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
19 fvex 5555 . . . . . . 7  |-  ( 1st `  g )  e.  _V
2019a1i 10 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  -> 
( 1st `  g
)  e.  _V )
21 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  C )  =  (
Base `  C )
22 eqid 2296 . . . . . . . . . . 11  |-  (  Hom  `  C )  =  (  Hom  `  C )
23 eqid 2296 . . . . . . . . . . 11  |-  (  Hom  `  D )  =  (  Hom  `  D )
243ad4antr 712 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  x  e.  ( Base `  A
) )  ->  (  Homf  `  C )  =  (  Homf 
`  D ) )
25 eqid 2296 . . . . . . . . . . . . 13  |-  ( Base `  A )  =  (
Base `  A )
26 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  r  =  ( 1st `  f
) )
27 relfunc 13752 . . . . . . . . . . . . . . 15  |-  Rel  ( A  Func  C )
28 simpllr 735 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )
2928simpld 445 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  f  e.  ( A  Func  C
) )
30 1st2ndbr 6185 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  f  e.  ( A  Func  C
) )  ->  ( 1st `  f ) ( A  Func  C )
( 2nd `  f
) )
3127, 29, 30sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  ( 1st `  f ) ( A  Func  C )
( 2nd `  f
) )
3226, 31eqbrtrd 4059 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  r
( A  Func  C
) ( 2nd `  f
) )
3325, 21, 32funcf1 13756 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  r : ( Base `  A
) --> ( Base `  C
) )
3433ffvelrnda 5681 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  x  e.  ( Base `  A
) )  ->  (
r `  x )  e.  ( Base `  C
) )
35 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  s  =  ( 1st `  g
) )
3628simprd 449 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  g  e.  ( A  Func  C
) )
37 1st2ndbr 6185 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  g  e.  ( A  Func  C
) )  ->  ( 1st `  g ) ( A  Func  C )
( 2nd `  g
) )
3827, 36, 37sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  ( 1st `  g ) ( A  Func  C )
( 2nd `  g
) )
3935, 38eqbrtrd 4059 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  s
( A  Func  C
) ( 2nd `  g
) )
4025, 21, 39funcf1 13756 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  s : ( Base `  A
) --> ( Base `  C
) )
4140ffvelrnda 5681 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  x  e.  ( Base `  A
) )  ->  (
s `  x )  e.  ( Base `  C
) )
4221, 22, 23, 24, 34, 41homfeqval 13616 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  x  e.  ( Base `  A
) )  ->  (
( r `  x
) (  Hom  `  C
) ( s `  x ) )  =  ( ( r `  x ) (  Hom  `  D ) ( s `
 x ) ) )
4342ixpeq2dva 6847 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  X_ x  e.  ( Base `  A
) ( ( r `
 x ) (  Hom  `  C )
( s `  x
) )  =  X_ x  e.  ( Base `  A ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) ) )
441homfeqbas 13615 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
4544ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  ( Base `  A )  =  ( Base `  B
) )
4645ixpeq1d 6844 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  X_ x  e.  ( Base `  A
) ( ( r `
 x ) (  Hom  `  D )
( s `  x
) )  =  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) ) )
4743, 46eqtrd 2328 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  X_ x  e.  ( Base `  A
) ( ( r `
 x ) (  Hom  `  C )
( s `  x
) )  =  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) ) )
48 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
r `  x )  =  ( r `  z ) )
49 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
s `  x )  =  ( s `  z ) )
5048, 49oveq12d 5892 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( r `  x
) (  Hom  `  C
) ( s `  x ) )  =  ( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )
5150cbvixpv 6850 . . . . . . . . . 10  |-  X_ x  e.  ( Base `  A
) ( ( r `
 x ) (  Hom  `  C )
( s `  x
) )  =  X_ z  e.  ( Base `  A ) ( ( r `  z ) (  Hom  `  C
) ( s `  z ) )
5251eleq2i 2360 . . . . . . . . 9  |-  ( a  e.  X_ x  e.  (
Base `  A )
( ( r `  x ) (  Hom  `  C ) ( s `
 x ) )  <-> 
a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )
5345adantr 451 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  ->  ( Base `  A )  =  (
Base `  B )
)
5453adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  ->  ( Base `  A )  =  ( Base `  B
) )
55 eqid 2296 . . . . . . . . . . . . 13  |-  (  Hom  `  A )  =  (  Hom  `  A )
56 eqid 2296 . . . . . . . . . . . . 13  |-  (  Hom  `  B )  =  (  Hom  `  B )
571ad6antr 716 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
(  Homf 
`  A )  =  (  Homf 
`  B ) )
58 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  ->  x  e.  ( Base `  A ) )
59 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
y  e.  ( Base `  A ) )
6025, 55, 56, 57, 58, 59homfeqval 13616 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( x (  Hom  `  A ) y )  =  ( x (  Hom  `  B )
y ) )
61 eqid 2296 . . . . . . . . . . . . . 14  |-  (comp `  C )  =  (comp `  C )
62 eqid 2296 . . . . . . . . . . . . . 14  |-  (comp `  D )  =  (comp `  D )
633ad7antr 718 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  (  Homf  `  C
)  =  (  Homf  `  D ) )
644ad7antr 718 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  (compf `  C )  =  (compf `  D ) )
6534adantlr 695 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  ->  (
r `  x )  e.  ( Base `  C
) )
6665ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( r `  x )  e.  (
Base `  C )
)
6733ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  ->  r : ( Base `  A
) --> ( Base `  C
) )
6867ffvelrnda 5681 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( r `  y
)  e.  ( Base `  C ) )
6968adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( r `  y )  e.  (
Base `  C )
)
7040ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  ->  s : ( Base `  A
) --> ( Base `  C
) )
7170ffvelrnda 5681 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( s `  y
)  e.  ( Base `  C ) )
7271adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( s `  y )  e.  (
Base `  C )
)
7332ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
r ( A  Func  C ) ( 2nd `  f
) )
7425, 55, 22, 73, 58, 59funcf2 13758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( x ( 2nd `  f ) y ) : ( x (  Hom  `  A )
y ) --> ( ( r `  x ) (  Hom  `  C
) ( r `  y ) ) )
7574ffvelrnda 5681 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( (
x ( 2nd `  f
) y ) `  h )  e.  ( ( r `  x
) (  Hom  `  C
) ( r `  y ) ) )
76 simplr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  ->  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )
77 fveq2 5541 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
r `  z )  =  ( r `  y ) )
78 fveq2 5541 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
s `  z )  =  ( s `  y ) )
7977, 78oveq12d 5892 . . . . . . . . . . . . . . . . 17  |-  ( z  =  y  ->  (
( r `  z
) (  Hom  `  C
) ( s `  z ) )  =  ( ( r `  y ) (  Hom  `  C ) ( s `
 y ) ) )
8079fvixp 6837 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) )  /\  y  e.  ( Base `  A
) )  ->  (
a `  y )  e.  ( ( r `  y ) (  Hom  `  C ) ( s `
 y ) ) )
8176, 80sylan 457 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( a `  y
)  e.  ( ( r `  y ) (  Hom  `  C
) ( s `  y ) ) )
8281adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( a `  y )  e.  ( ( r `  y
) (  Hom  `  C
) ( s `  y ) ) )
8321, 22, 61, 62, 63, 64, 66, 69, 72, 75, 82comfeqval 13627 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( (
a `  y )
( <. ( r `  x ) ,  ( r `  y )
>. (comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) ) )
8441adantlr 695 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  ->  (
s `  x )  e.  ( Base `  C
) )
8584ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( s `  x )  e.  (
Base `  C )
)
86 fveq2 5541 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  x  ->  (
r `  z )  =  ( r `  x ) )
87 fveq2 5541 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  x  ->  (
s `  z )  =  ( s `  x ) )
8886, 87oveq12d 5892 . . . . . . . . . . . . . . . . 17  |-  ( z  =  x  ->  (
( r `  z
) (  Hom  `  C
) ( s `  z ) )  =  ( ( r `  x ) (  Hom  `  C ) ( s `
 x ) ) )
8988fvixp 6837 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) )  /\  x  e.  ( Base `  A
) )  ->  (
a `  x )  e.  ( ( r `  x ) (  Hom  `  C ) ( s `
 x ) ) )
9089adantll 694 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  ->  (
a `  x )  e.  ( ( r `  x ) (  Hom  `  C ) ( s `
 x ) ) )
9190ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( a `  x )  e.  ( ( r `  x
) (  Hom  `  C
) ( s `  x ) ) )
9239ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
s ( A  Func  C ) ( 2nd `  g
) )
9325, 55, 22, 92, 58, 59funcf2 13758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( x ( 2nd `  g ) y ) : ( x (  Hom  `  A )
y ) --> ( ( s `  x ) (  Hom  `  C
) ( s `  y ) ) )
9493ffvelrnda 5681 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( (
x ( 2nd `  g
) y ) `  h )  e.  ( ( s `  x
) (  Hom  `  C
) ( s `  y ) ) )
9521, 22, 61, 62, 63, 64, 66, 85, 72, 91, 94comfeqval 13627 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( (
( x ( 2nd `  g ) y ) `
 h ) (
<. ( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) )
9683, 95eqeq12d 2310 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  /\  y  e.  ( Base `  A
) )  /\  h  e.  ( x (  Hom  `  A ) y ) )  ->  ( (
( a `  y
) ( <. (
r `  x ) ,  ( r `  y ) >. (comp `  C ) ( s `
 y ) ) ( ( x ( 2nd `  f ) y ) `  h
) )  =  ( ( ( x ( 2nd `  g ) y ) `  h
) ( <. (
r `  x ) ,  ( s `  x ) >. (comp `  C ) ( s `
 y ) ) ( a `  x
) )  <->  ( (
a `  y )
( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) ) )
9760, 96raleqbidva 2763 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  /\  s  =  ( 1st `  g ) )  /\  a  e.  X_ z  e.  ( Base `  A
) ( ( r `
 z ) (  Hom  `  C )
( s `  z
) ) )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( A. h  e.  ( x (  Hom  `  A ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) )  <->  A. h  e.  (
x (  Hom  `  B
) y ) ( ( a `  y
) ( <. (
r `  x ) ,  ( r `  y ) >. (comp `  D ) ( s `
 y ) ) ( ( x ( 2nd `  f ) y ) `  h
) )  =  ( ( ( x ( 2nd `  g ) y ) `  h
) ( <. (
r `  x ) ,  ( s `  x ) >. (comp `  D ) ( s `
 y ) ) ( a `  x
) ) ) )
9854, 97raleqbidva 2763 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  /\  x  e.  ( Base `  A
) )  ->  ( A. y  e.  ( Base `  A ) A. h  e.  ( x
(  Hom  `  A ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) )  <->  A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) ) )
9953, 98raleqbidva 2763 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ z  e.  (
Base `  A )
( ( r `  z ) (  Hom  `  C ) ( s `
 z ) ) )  ->  ( A. x  e.  ( Base `  A ) A. y  e.  ( Base `  A
) A. h  e.  ( x (  Hom  `  A ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) )  <->  A. x  e.  ( Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x (  Hom  `  B ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) ) )
10052, 99sylan2b 461 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  a  e.  X_ x  e.  (
Base `  A )
( ( r `  x ) (  Hom  `  C ) ( s `
 x ) ) )  ->  ( A. x  e.  ( Base `  A ) A. y  e.  ( Base `  A
) A. h  e.  ( x (  Hom  `  A ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) )  <->  A. x  e.  ( Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x (  Hom  `  B ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) ) )
10147, 100rabeqbidva 2797 . . . . . . 7  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  { a  e.  X_ x  e.  (
Base `  A )
( ( r `  x ) (  Hom  `  C ) ( s `
 x ) )  |  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) A. h  e.  ( x (  Hom  `  A ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) ) }  =  { a  e.  X_ x  e.  (
Base `  B )
( ( r `  x ) (  Hom  `  D ) ( s `
 x ) )  |  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) A. h  e.  ( x (  Hom  `  B ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
102 csbeq1a 3102 . . . . . . . 8  |-  ( s  =  ( 1st `  g
)  ->  { a  e.  X_ x  e.  (
Base `  B )
( ( r `  x ) (  Hom  `  D ) ( s `
 x ) )  |  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) A. h  e.  ( x (  Hom  `  B ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) }  =  [_ ( 1st `  g )  / 
s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x
) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
103102adantl 452 . . . . . . 7  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  { a  e.  X_ x  e.  (
Base `  B )
( ( r `  x ) (  Hom  `  D ) ( s `
 x ) )  |  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) A. h  e.  ( x (  Hom  `  B ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) }  =  [_ ( 1st `  g )  / 
s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x
) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
104101, 103eqtrd 2328 . . . . . 6  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  { a  e.  X_ x  e.  (
Base `  A )
( ( r `  x ) (  Hom  `  C ) ( s `
 x ) )  |  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) A. h  e.  ( x (  Hom  `  A ) y ) ( ( a `  y ) ( <.
( r `  x
) ,  ( r `
 y ) >.
(comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) ) }  =  [_ ( 1st `  g )  / 
s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x
) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
10516, 18, 20, 104csbiedf 3131 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  ->  [_ ( 1st `  g
)  /  s ]_ { a  e.  X_ x  e.  ( Base `  A ) ( ( r `  x ) (  Hom  `  C
) ( s `  x ) )  | 
A. x  e.  (
Base `  A ) A. y  e.  ( Base `  A ) A. h  e.  ( x
(  Hom  `  A ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) ) }  =  [_ ( 1st `  g )  / 
s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x
) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
106 csbeq1a 3102 . . . . . 6  |-  ( r  =  ( 1st `  f
)  ->  [_ ( 1st `  g )  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) }  =  [_ ( 1st `  f )  / 
r ]_ [_ ( 1st `  g )  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
107106adantl 452 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  ->  [_ ( 1st `  g
)  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) }  =  [_ ( 1st `  f )  / 
r ]_ [_ ( 1st `  g )  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
108105, 107eqtrd 2328 . . . 4  |-  ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  ->  [_ ( 1st `  g
)  /  s ]_ { a  e.  X_ x  e.  ( Base `  A ) ( ( r `  x ) (  Hom  `  C
) ( s `  x ) )  | 
A. x  e.  (
Base `  A ) A. y  e.  ( Base `  A ) A. h  e.  ( x
(  Hom  `  A ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) ) }  =  [_ ( 1st `  f )  / 
r ]_ [_ ( 1st `  g )  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
10911, 13, 15, 108csbiedf 3131 . . 3  |-  ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  ->  [_ ( 1st `  f )  / 
r ]_ [_ ( 1st `  g )  /  s ]_ { a  e.  X_ x  e.  ( Base `  A ) ( ( r `  x ) (  Hom  `  C
) ( s `  x ) )  | 
A. x  e.  (
Base `  A ) A. y  e.  ( Base `  A ) A. h  e.  ( x
(  Hom  `  A ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) ) }  =  [_ ( 1st `  f )  / 
r ]_ [_ ( 1st `  g )  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
1109, 10, 109mpt2eq123dva 5925 . 2  |-  ( ph  ->  ( f  e.  ( A  Func  C ) ,  g  e.  ( A  Func  C )  |->  [_ ( 1st `  f )  /  r ]_ [_ ( 1st `  g )  / 
s ]_ { a  e.  X_ x  e.  ( Base `  A ) ( ( r `  x
) (  Hom  `  C
) ( s `  x ) )  | 
A. x  e.  (
Base `  A ) A. y  e.  ( Base `  A ) A. h  e.  ( x
(  Hom  `  A ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) ) } )  =  ( f  e.  ( B 
Func  D ) ,  g  e.  ( B  Func  D )  |->  [_ ( 1st `  f
)  /  r ]_ [_ ( 1st `  g
)  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } ) )
111 eqid 2296 . . 3  |-  ( A Nat 
C )  =  ( A Nat  C )
112111, 25, 55, 22, 61natfval 13836 . 2  |-  ( A Nat 
C )  =  ( f  e.  ( A 
Func  C ) ,  g  e.  ( A  Func  C )  |->  [_ ( 1st `  f
)  /  r ]_ [_ ( 1st `  g
)  /  s ]_ { a  e.  X_ x  e.  ( Base `  A ) ( ( r `  x ) (  Hom  `  C
) ( s `  x ) )  | 
A. x  e.  (
Base `  A ) A. y  e.  ( Base `  A ) A. h  e.  ( x
(  Hom  `  A ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  C )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  C )
( s `  y
) ) ( a `
 x ) ) } )
113 eqid 2296 . . 3  |-  ( B Nat 
D )  =  ( B Nat  D )
114 eqid 2296 . . 3  |-  ( Base `  B )  =  (
Base `  B )
115113, 114, 56, 23, 62natfval 13836 . 2  |-  ( B Nat 
D )  =  ( f  e.  ( B 
Func  D ) ,  g  e.  ( B  Func  D )  |->  [_ ( 1st `  f
)  /  r ]_ [_ ( 1st `  g
)  /  s ]_ { a  e.  X_ x  e.  ( Base `  B ) ( ( r `  x ) (  Hom  `  D
) ( s `  x ) )  | 
A. x  e.  (
Base `  B ) A. y  e.  ( Base `  B ) A. h  e.  ( x
(  Hom  `  B ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y )
>. (comp `  D )
( s `  y
) ) ( ( x ( 2nd `  f
) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
) y ) `  h ) ( <.
( r `  x
) ,  ( s `
 x ) >.
(comp `  D )
( s `  y
) ) ( a `
 x ) ) } )
116110, 112, 1153eqtr4g 2353 1  |-  ( ph  ->  ( A Nat  C )  =  ( B Nat  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   F/_wnfc 2419   A.wral 2556   {crab 2560   _Vcvv 2801   [_csb 3094   <.cop 3656   class class class wbr 4039   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   X_cixp 6833   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582    Homf chomf 13584  compfccomf 13585    Func cfunc 13744   Nat cnat 13831
This theorem is referenced by:  fucpropd  13867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-map 6790  df-ixp 6834  df-cat 13586  df-cid 13587  df-homf 13588  df-comf 13589  df-func 13748  df-nat 13833
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