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Theorem natpropd 15324
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  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
fucpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
fucpropd.3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  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  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
2 fucpropd.2 . . . 4  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
3 fucpropd.3 . . . 4  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  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 15248 . . 3  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
109adantr 465 . . 3  |-  ( (
ph  /\  f  e.  ( A  Func  C ) )  ->  ( A  Func  C )  =  ( B  Func  D )
)
11 nfv 1694 . . . 4  |-  F/ r ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )
12 nfcsb1v 3436 . . . . 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 11 . . . 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 5866 . . . . 5  |-  ( 1st `  f )  e.  _V
1514a1i 11 . . . 4  |-  ( (
ph  /\  ( f  e.  ( A  Func  C
)  /\  g  e.  ( A  Func  C ) ) )  ->  ( 1st `  f )  e. 
_V )
16 nfv 1694 . . . . . 6  |-  F/ s ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )
17 nfcsb1v 3436 . . . . . . 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 11 . . . . . 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 5866 . . . . . . 7  |-  ( 1st `  g )  e.  _V
2019a1i 11 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )  /\  r  =  ( 1st `  f ) )  -> 
( 1st `  g
)  e.  _V )
21 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  C )  =  (
Base `  C )
22 eqid 2443 . . . . . . . . . . 11  |-  ( Hom  `  C )  =  ( Hom  `  C )
23 eqid 2443 . . . . . . . . . . 11  |-  ( Hom  `  D )  =  ( Hom  `  D )
243ad4antr 731 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  /\  x  e.  ( Base `  A
) )  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
25 eqid 2443 . . . . . . . . . . . . 13  |-  ( Base `  A )  =  (
Base `  A )
26 simplr 755 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  r  =  ( 1st `  f
) )
27 relfunc 15210 . . . . . . . . . . . . . . 15  |-  Rel  ( A  Func  C )
28 simpllr 760 . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  f  e.  ( A  Func  C
) )  ->  ( 1st `  f ) ( A  Func  C )
( 2nd `  f
) )
3127, 29, 30sylancr 663 . . . . . . . . . . . . . 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 4457 . . . . . . . . . . . . 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 15214 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  r : ( Base `  A
) --> ( Base `  C
) )
3433ffvelrnda 6016 . . . . . . . . . . 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 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  s  =  ( 1st `  g
) )
3628simprd 463 . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  g  e.  ( A  Func  C
) )  ->  ( 1st `  g ) ( A  Func  C )
( 2nd `  g
) )
3827, 36, 37sylancr 663 . . . . . . . . . . . . . 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 4457 . . . . . . . . . . . . 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 15214 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  s : ( Base `  A
) --> ( Base `  C
) )
4140ffvelrnda 6016 . . . . . . . . . . 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 15074 . . . . . . . . . 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 7486 . . . . . . . . 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 15073 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
4544ad3antrrr 729 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  ( Base `  A )  =  ( Base `  B
) )
4645ixpeq1d 7483 . . . . . . . . 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 2484 . . . . . . . 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 5856 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
r `  x )  =  ( r `  z ) )
49 fveq2 5856 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
s `  x )  =  ( s `  z ) )
5048, 49oveq12d 6299 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( r `  x
) ( Hom  `  C
) ( s `  x ) )  =  ( ( r `  z ) ( Hom  `  C ) ( s `
 z ) ) )
5150cbvixpv 7489 . . . . . . . . . 10  |-  X_ x  e.  ( Base `  A
) ( ( r `
 x ) ( Hom  `  C )
( s `  x
) )  =  X_ z  e.  ( Base `  A ) ( ( r `  z ) ( Hom  `  C
) ( s `  z ) )
5251eleq2i 2521 . . . . . . . . 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 465 . . . . . . . . . 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 465 . . . . . . . . . . 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 2443 . . . . . . . . . . . . 13  |-  ( Hom  `  A )  =  ( Hom  `  A )
56 eqid 2443 . . . . . . . . . . . . 13  |-  ( Hom  `  B )  =  ( Hom  `  B )
571ad6antr 735 . . . . . . . . . . . . 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 ) )  -> 
( Hom f  `  A )  =  ( Hom f  `  B ) )
58 simplr 755 . . . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 15074 . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . 14  |-  (comp `  C )  =  (comp `  C )
62 eqid 2443 . . . . . . . . . . . . . 14  |-  (comp `  D )  =  (comp `  D )
633ad7antr 737 . . . . . . . . . . . . . 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 ) )  ->  ( Hom f  `  C
)  =  ( Hom f  `  D ) )
644ad7antr 737 . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . 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 6016 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . 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 6016 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . 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 15216 . . . . . . . . . . . . . . 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 6016 . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 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 5856 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
r `  z )  =  ( r `  y ) )
78 fveq2 5856 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
s `  z )  =  ( s `  y ) )
7977, 78oveq12d 6299 . . . . . . . . . . . . . . . . 17  |-  ( z  =  y  ->  (
( r `  z
) ( Hom  `  C
) ( s `  z ) )  =  ( ( r `  y ) ( Hom  `  C ) ( s `
 y ) ) )
8079fvixp 7476 . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 15085 . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 5856 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  x  ->  (
r `  z )  =  ( r `  x ) )
87 fveq2 5856 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  x  ->  (
s `  z )  =  ( s `  x ) )
8886, 87oveq12d 6299 . . . . . . . . . . . . . . . . 17  |-  ( z  =  x  ->  (
( r `  z
) ( Hom  `  C
) ( s `  z ) )  =  ( ( r `  x ) ( Hom  `  C ) ( s `
 x ) ) )
8988fvixp 7476 . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . 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 15216 . . . . . . . . . . . . . . 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 6016 . . . . . . . . . . . . . 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 15085 . . . . . . . . . . . . 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 2465 . . . . . . . . . . . 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 3056 . . . . . . . . . . 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 3056 . . . . . . . . . 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 3056 . . . . . . . . 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 475 . . . . . . . 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 3091 . . . . . . 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 3429 . . . . . . . 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 466 . . . . . . 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 2484 . . . . . 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 3441 . . . . 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 3429 . . . . . 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 466 . . . . 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 2484 . . . 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 3441 . . 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 6343 . 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 2443 . . 3  |-  ( A Nat 
C )  =  ( A Nat  C )
112111, 25, 55, 22, 61natfval 15294 . 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 2443 . . 3  |-  ( B Nat 
D )  =  ( B Nat  D )
114 eqid 2443 . . 3  |-  ( Base `  B )  =  (
Base `  B )
115113, 114, 56, 23, 62natfval 15294 . 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 2509 1  |-  ( ph  ->  ( A Nat  C )  =  ( B Nat  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   F/_wnfc 2591   A.wral 2793   {crab 2797   _Vcvv 3095   [_csb 3420   <.cop 4020   class class class wbr 4437   Rel wrel 4994   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   1stc1st 6783   2ndc2nd 6784   X_cixp 7471   Basecbs 14614   Hom chom 14690  compcco 14691   Catccat 15043   Hom f chomf 15045  compfccomf 15046    Func cfunc 15202   Nat cnat 15289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-map 7424  df-ixp 7472  df-cat 15047  df-cid 15048  df-homf 15049  df-comf 15050  df-func 15206  df-nat 15291
This theorem is referenced by:  fucpropd  15325
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