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Theorem hofpropd 14365
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
hofpropd.1  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
hofpropd.2  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
hofpropd.c  |-  ( ph  ->  C  e.  Cat )
hofpropd.d  |-  ( ph  ->  D  e.  Cat )
Assertion
Ref Expression
hofpropd  |-  ( ph  ->  (HomF
`  C )  =  (HomF
`  D ) )

Proof of Theorem hofpropd
Dummy variables  f 
g  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofpropd.1 . . 3  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
21homfeqbas 13923 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
32, 2xpeq12d 4904 . . . 4  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
43adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) )  ->  ( ( Base `  C )  X.  ( Base `  C ) )  =  ( ( Base `  D )  X.  ( Base `  D ) ) )
5 eqid 2437 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
6 eqid 2437 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
7 eqid 2437 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
81adantr 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  (  Homf  `  C )  =  (  Homf  `  D ) )
9 xp1st 6377 . . . . . . 7  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 1st `  y
)  e.  ( Base `  C ) )
109ad2antll 711 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 1st `  y
)  e.  ( Base `  C ) )
11 xp1st 6377 . . . . . . 7  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
1211ad2antrl 710 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
135, 6, 7, 8, 10, 12homfeqval 13924 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  =  ( ( 1st `  y
) (  Hom  `  D
) ( 1st `  x
) ) )
14 xp2nd 6378 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 2nd `  x
)  e.  ( Base `  C ) )
1514ad2antrl 710 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  C ) )
16 xp2nd 6378 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 2nd `  y
)  e.  ( Base `  C ) )
1716ad2antll 711 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 2nd `  y
)  e.  ( Base `  C ) )
185, 6, 7, 8, 15, 17homfeqval 13924 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( ( 2nd `  x ) (  Hom  `  C ) ( 2nd `  y ) )  =  ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) )
1918adantr 453 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( (
Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  f  e.  ( ( 1st `  y
) (  Hom  `  C
) ( 1st `  x
) ) )  -> 
( ( 2nd `  x
) (  Hom  `  C
) ( 2nd `  y
) )  =  ( ( 2nd `  x
) (  Hom  `  D
) ( 2nd `  y
) ) )
205, 6, 7, 8, 12, 15homfeqval 13924 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( ( 1st `  x ) (  Hom  `  C ) ( 2nd `  x ) )  =  ( ( 1st `  x
) (  Hom  `  D
) ( 2nd `  x
) ) )
21 df-ov 6085 . . . . . . . . 9  |-  ( ( 1st `  x ) (  Hom  `  C
) ( 2nd `  x
) )  =  ( (  Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
22 df-ov 6085 . . . . . . . . 9  |-  ( ( 1st `  x ) (  Hom  `  D
) ( 2nd `  x
) )  =  ( (  Hom  `  D
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
2320, 21, 223eqtr3g 2492 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( (  Hom  `  C ) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  =  ( (  Hom  `  D
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
24 1st2nd2 6387 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
2524ad2antrl 710 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
2625fveq2d 5733 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( (  Hom  `  C ) `  x
)  =  ( (  Hom  `  C ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2725fveq2d 5733 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( (  Hom  `  D ) `  x
)  =  ( (  Hom  `  D ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2823, 26, 273eqtr4d 2479 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( (  Hom  `  C ) `  x
)  =  ( (  Hom  `  D ) `  x ) )
2928adantr 453 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( (
Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  ->  ( (  Hom  `  C ) `  x
)  =  ( (  Hom  `  D ) `  x ) )
30 eqid 2437 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
31 eqid 2437 . . . . . . . 8  |-  (comp `  D )  =  (comp `  D )
328ad2antrr 708 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  (  Homf  `  C
)  =  (  Homf  `  D ) )
33 hofpropd.2 . . . . . . . . 9  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
3433ad3antrrr 712 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  (compf `  C )  =  (compf `  D ) )
3510ad2antrr 708 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( 1st `  y )  e.  (
Base `  C )
)
3612ad2antrr 708 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( 1st `  x )  e.  (
Base `  C )
)
3717ad2antrr 708 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( 2nd `  y )  e.  (
Base `  C )
)
38 simplrl 738 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  f  e.  ( ( 1st `  y
) (  Hom  `  C
) ( 1st `  x
) ) )
3925ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
4039oveq1d 6097 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( x
(comp `  C )
( 2nd `  y
) )  =  (
<. ( 1st `  x
) ,  ( 2nd `  x ) >. (comp `  C ) ( 2nd `  y ) ) )
4140oveqd 6099 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( g
( x (comp `  C ) ( 2nd `  y ) ) h )  =  ( g ( <. ( 1st `  x
) ,  ( 2nd `  x ) >. (comp `  C ) ( 2nd `  y ) ) h ) )
42 hofpropd.c . . . . . . . . . . 11  |-  ( ph  ->  C  e.  Cat )
4342ad3antrrr 712 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  C  e.  Cat )
4415ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( 2nd `  x )  e.  (
Base `  C )
)
4526adantr 453 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( (
Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  ->  ( (  Hom  `  C ) `  x
)  =  ( (  Hom  `  C ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
4645, 21syl6eqr 2487 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( (
Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  ->  ( (  Hom  `  C ) `  x
)  =  ( ( 1st `  x ) (  Hom  `  C
) ( 2nd `  x
) ) )
4746eleq2d 2504 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( (
Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  ->  ( h  e.  ( (  Hom  `  C
) `  x )  <->  h  e.  ( ( 1st `  x ) (  Hom  `  C ) ( 2nd `  x ) ) ) )
4847biimpa 472 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  h  e.  ( ( 1st `  x
) (  Hom  `  C
) ( 2nd `  x
) ) )
49 simplrr 739 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  g  e.  ( ( 2nd `  x
) (  Hom  `  C
) ( 2nd `  y
) ) )
505, 6, 30, 43, 36, 44, 37, 48, 49catcocl 13911 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( g
( <. ( 1st `  x
) ,  ( 2nd `  x ) >. (comp `  C ) ( 2nd `  y ) ) h )  e.  ( ( 1st `  x ) (  Hom  `  C
) ( 2nd `  y
) ) )
5141, 50eqeltrd 2511 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( g
( x (comp `  C ) ( 2nd `  y ) ) h )  e.  ( ( 1st `  x ) (  Hom  `  C
) ( 2nd `  y
) ) )
525, 6, 30, 31, 32, 34, 35, 36, 37, 38, 51comfeqval 13935 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( (
g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  C
) ( 2nd `  y
) ) f )  =  ( ( g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  D
) ( 2nd `  y
) ) f ) )
535, 6, 30, 31, 32, 34, 36, 44, 37, 48, 49comfeqval 13935 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( g
( <. ( 1st `  x
) ,  ( 2nd `  x ) >. (comp `  C ) ( 2nd `  y ) ) h )  =  ( g ( <. ( 1st `  x
) ,  ( 2nd `  x ) >. (comp `  D ) ( 2nd `  y ) ) h ) )
5439oveq1d 6097 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( x
(comp `  D )
( 2nd `  y
) )  =  (
<. ( 1st `  x
) ,  ( 2nd `  x ) >. (comp `  D ) ( 2nd `  y ) ) )
5554oveqd 6099 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( g
( x (comp `  D ) ( 2nd `  y ) ) h )  =  ( g ( <. ( 1st `  x
) ,  ( 2nd `  x ) >. (comp `  D ) ( 2nd `  y ) ) h ) )
5653, 41, 553eqtr4d 2479 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( g
( x (comp `  C ) ( 2nd `  y ) ) h )  =  ( g ( x (comp `  D ) ( 2nd `  y ) ) h ) )
5756oveq1d 6097 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( (
g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  D
) ( 2nd `  y
) ) f )  =  ( ( g ( x (comp `  D ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  D
) ( 2nd `  y
) ) f ) )
5852, 57eqtrd 2469 . . . . . 6  |-  ( ( ( ( ph  /\  ( x  e.  (
( Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  /\  h  e.  ( (  Hom  `  C
) `  x )
)  ->  ( (
g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  C
) ( 2nd `  y
) ) f )  =  ( ( g ( x (comp `  D ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  D
) ( 2nd `  y
) ) f ) )
5929, 58mpteq12dva 4287 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( (
Base `  C )  X.  ( Base `  C
) )  /\  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ) )  /\  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) )  /\  g  e.  ( ( 2nd `  x ) (  Hom  `  C )
( 2nd `  y
) ) ) )  ->  ( h  e.  ( (  Hom  `  C
) `  x )  |->  ( ( g ( x (comp `  C
) ( 2nd `  y
) ) h ) ( <. ( 1st `  y
) ,  ( 1st `  x ) >. (comp `  C ) ( 2nd `  y ) ) f ) )  =  ( h  e.  ( (  Hom  `  D ) `  x )  |->  ( ( g ( x (comp `  D ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  D
) ( 2nd `  y
) ) f ) ) )
6013, 19, 59mpt2eq123dva 6136 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( f  e.  ( ( 1st `  y
) (  Hom  `  C
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  C ) ( 2nd `  y ) )  |->  ( h  e.  ( (  Hom  `  C ) `  x )  |->  ( ( g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  C
) ( 2nd `  y
) ) f ) ) )  =  ( f  e.  ( ( 1st `  y ) (  Hom  `  D
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  D ) ( 2nd `  y ) )  |->  ( h  e.  ( (  Hom  `  D ) `  x )  |->  ( ( g ( x (comp `  D ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  D
) ( 2nd `  y
) ) f ) ) ) )
613, 4, 60mpt2eq123dva 6136 . . 3  |-  ( ph  ->  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  C
) )  |->  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  C
) ( 2nd `  y
) )  |->  ( h  e.  ( (  Hom  `  C ) `  x
)  |->  ( ( g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  C
) ( 2nd `  y
) ) f ) ) ) )  =  ( x  e.  ( ( Base `  D
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  D )  X.  ( Base `  D
) )  |->  ( f  e.  ( ( 1st `  y ) (  Hom  `  D ) ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  D
) ( 2nd `  y
) )  |->  ( h  e.  ( (  Hom  `  D ) `  x
)  |->  ( ( g ( x (comp `  D ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  D
) ( 2nd `  y
) ) f ) ) ) ) )
621, 61opeq12d 3993 . 2  |-  ( ph  -> 
<. (  Homf 
`  C ) ,  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  C
) )  |->  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  C
) ( 2nd `  y
) )  |->  ( h  e.  ( (  Hom  `  C ) `  x
)  |->  ( ( g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  C
) ( 2nd `  y
) ) f ) ) ) ) >.  =  <. (  Homf  `  D ) ,  ( x  e.  ( ( Base `  D
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  D )  X.  ( Base `  D
) )  |->  ( f  e.  ( ( 1st `  y ) (  Hom  `  D ) ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  D
) ( 2nd `  y
) )  |->  ( h  e.  ( (  Hom  `  D ) `  x
)  |->  ( ( g ( x (comp `  D ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  D
) ( 2nd `  y
) ) f ) ) ) ) >.
)
63 eqid 2437 . . 3  |-  (HomF `  C
)  =  (HomF `  C
)
6463, 42, 5, 6, 30hofval 14350 . 2  |-  ( ph  ->  (HomF
`  C )  = 
<. (  Homf 
`  C ) ,  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  C
) )  |->  ( f  e.  ( ( 1st `  y ) (  Hom  `  C ) ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  C
) ( 2nd `  y
) )  |->  ( h  e.  ( (  Hom  `  C ) `  x
)  |->  ( ( g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  C
) ( 2nd `  y
) ) f ) ) ) ) >.
)
65 eqid 2437 . . 3  |-  (HomF `  D
)  =  (HomF `  D
)
66 hofpropd.d . . 3  |-  ( ph  ->  D  e.  Cat )
67 eqid 2437 . . 3  |-  ( Base `  D )  =  (
Base `  D )
6865, 66, 67, 7, 31hofval 14350 . 2  |-  ( ph  ->  (HomF
`  D )  = 
<. (  Homf 
`  D ) ,  ( x  e.  ( ( Base `  D
)  X.  ( Base `  D ) ) ,  y  e.  ( (
Base `  D )  X.  ( Base `  D
) )  |->  ( f  e.  ( ( 1st `  y ) (  Hom  `  D ) ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  D
) ( 2nd `  y
) )  |->  ( h  e.  ( (  Hom  `  D ) `  x
)  |->  ( ( g ( x (comp `  D ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  D
) ( 2nd `  y
) ) f ) ) ) ) >.
)
6962, 64, 683eqtr4d 2479 1  |-  ( ph  ->  (HomF
`  C )  =  (HomF
`  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3818    e. cmpt 4267    X. cxp 4877   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   1stc1st 6348   2ndc2nd 6349   Basecbs 13470    Hom chom 13541  compcco 13542   Catccat 13890    Homf chomf 13892  compfccomf 13893  HomFchof 14346
This theorem is referenced by:  yonpropd  14366  oppcyon  14367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-cat 13894  df-homf 13896  df-comf 13897  df-hof 14348
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