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Theorem natpropd 15203
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 15127 . . 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 1683 . . . 4  |-  F/ r ( ph  /\  (
f  e.  ( A 
Func  C )  /\  g  e.  ( A  Func  C
) ) )
12 nfcsb1v 3451 . . . . 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 5876 . . . . 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 1683 . . . . . 6  |-  F/ s ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )
17 nfcsb1v 3451 . . . . . . 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 5876 . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( Base `  C )  =  (
Base `  C )
22 eqid 2467 . . . . . . . . . . 11  |-  ( Hom  `  C )  =  ( Hom  `  C )
23 eqid 2467 . . . . . . . . . . 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 2467 . . . . . . . . . . . . 13  |-  ( Base `  A )  =  (
Base `  A )
26 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  r  =  ( 1st `  f
) )
27 relfunc 15089 . . . . . . . . . . . . . . 15  |-  Rel  ( A  Func  C )
28 simpllr 758 . . . . . . . . . . . . . . . 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 6833 . . . . . . . . . . . . . . 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 4467 . . . . . . . . . . . . 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 15093 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  r : ( Base `  A
) --> ( Base `  C
) )
3433ffvelrnda 6021 . . . . . . . . . . 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 6833 . . . . . . . . . . . . . . 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 4467 . . . . . . . . . . . . 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 15093 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( f  e.  ( A  Func  C )  /\  g  e.  ( A  Func  C ) ) )  /\  r  =  ( 1st `  f
) )  /\  s  =  ( 1st `  g
) )  ->  s : ( Base `  A
) --> ( Base `  C
) )
4140ffvelrnda 6021 . . . . . . . . . . 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 14953 . . . . . . . . . 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 7484 . . . . . . . . 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 14952 . . . . . . . . . . 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 7481 . . . . . . . . 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 2508 . . . . . . . 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 5866 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
r `  x )  =  ( r `  z ) )
49 fveq2 5866 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
s `  x )  =  ( s `  z ) )
5048, 49oveq12d 6302 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( r `  x
) ( Hom  `  C
) ( s `  x ) )  =  ( ( r `  z ) ( Hom  `  C ) ( s `
 z ) ) )
5150cbvixpv 7487 . . . . . . . . . 10  |-  X_ x  e.  ( Base `  A
) ( ( r `
 x ) ( Hom  `  C )
( s `  x
) )  =  X_ z  e.  ( Base `  A ) ( ( r `  z ) ( Hom  `  C
) ( s `  z ) )
5251eleq2i 2545 . . . . . . . . 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 2467 . . . . . . . . . . . . 13  |-  ( Hom  `  A )  =  ( Hom  `  A )
56 eqid 2467 . . . . . . . . . . . . 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 754 . . . . . . . . . . . . 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 14953 . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  (comp `  C )  =  (comp `  C )
62 eqid 2467 . . . . . . . . . . . . . 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 6021 . . . . . . . . . . . . . . 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 6021 . . . . . . . . . . . . . . 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 15095 . . . . . . . . . . . . . . 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 6021 . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
r `  z )  =  ( r `  y ) )
78 fveq2 5866 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
s `  z )  =  ( s `  y ) )
7977, 78oveq12d 6302 . . . . . . . . . . . . . . . . 17  |-  ( z  =  y  ->  (
( r `  z
) ( Hom  `  C
) ( s `  z ) )  =  ( ( r `  y ) ( Hom  `  C ) ( s `
 y ) ) )
8079fvixp 7474 . . . . . . . . . . . . . . . 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 14964 . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  x  ->  (
r `  z )  =  ( r `  x ) )
87 fveq2 5866 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  x  ->  (
s `  z )  =  ( s `  x ) )
8886, 87oveq12d 6302 . . . . . . . . . . . . . . . . 17  |-  ( z  =  x  ->  (
( r `  z
) ( Hom  `  C
) ( s `  z ) )  =  ( ( r `  x ) ( Hom  `  C ) ( s `
 x ) ) )
8988fvixp 7474 . . . . . . . . . . . . . . . 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 15095 . . . . . . . . . . . . . . 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 6021 . . . . . . . . . . . . . 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 14964 . . . . . . . . . . . . 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 2489 . . . . . . . . . . . 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 3074 . . . . . . . . . . 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 3074 . . . . . . . . . 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 3074 . . . . . . . . 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 3109 . . . . . . 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 3444 . . . . . . . 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 2508 . . . . . 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 3456 . . . . 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 3444 . . . . . 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 2508 . . . 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 3456 . . 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 6342 . 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 2467 . . 3  |-  ( A Nat 
C )  =  ( A Nat  C )
112111, 25, 55, 22, 61natfval 15173 . 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 2467 . . 3  |-  ( B Nat 
D )  =  ( B Nat  D )
114 eqid 2467 . . 3  |-  ( Base `  B )  =  (
Base `  B )
115113, 114, 56, 23, 62natfval 15173 . 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 2533 1  |-  ( ph  ->  ( A Nat  C )  =  ( B Nat  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   F/_wnfc 2615   A.wral 2814   {crab 2818   _Vcvv 3113   [_csb 3435   <.cop 4033   class class class wbr 4447   Rel wrel 5004   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1stc1st 6782   2ndc2nd 6783   X_cixp 7469   Basecbs 14490   Hom chom 14566  compcco 14567   Catccat 14919   Hom f chomf 14921  compfccomf 14922    Func cfunc 15081   Nat cnat 15168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422  df-ixp 7470  df-cat 14923  df-cid 14924  df-homf 14925  df-comf 14926  df-func 15085  df-nat 15170
This theorem is referenced by:  fucpropd  15204
  Copyright terms: Public domain W3C validator