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Theorem hofpropd 14090
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 13648 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
32, 2xpeq12d 4751 . . . 4  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
43adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) )  ->  ( ( Base `  C )  X.  ( Base `  C ) )  =  ( ( Base `  D )  X.  ( Base `  D ) ) )
5 eqid 2316 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
6 eqid 2316 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
7 eqid 2316 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
81adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  (  Homf  `  C )  =  (  Homf  `  D ) )
9 xp1st 6191 . . . . . . 7  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 1st `  y
)  e.  ( Base `  C ) )
109ad2antll 709 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 1st `  y
)  e.  ( Base `  C ) )
11 xp1st 6191 . . . . . . 7  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
1211ad2antrl 708 . . . . . 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 13649 . . . . 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 6192 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 2nd `  x
)  e.  ( Base `  C ) )
1514ad2antrl 708 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  C ) )
16 xp2nd 6192 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 2nd `  y
)  e.  ( Base `  C ) )
1716ad2antll 709 . . . . . . 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 13649 . . . . . 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 451 . . . . 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 13649 . . . . . . . . 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 5903 . . . . . . . . 9  |-  ( ( 1st `  x ) (  Hom  `  C
) ( 2nd `  x
) )  =  ( (  Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
22 df-ov 5903 . . . . . . . . 9  |-  ( ( 1st `  x ) (  Hom  `  D
) ( 2nd `  x
) )  =  ( (  Hom  `  D
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
2320, 21, 223eqtr3g 2371 . . . . . . . 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 6201 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
2524ad2antrl 708 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
2625fveq2d 5567 . . . . . . . 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 5567 . . . . . . . 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 2358 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( (  Hom  `  C ) `  x
)  =  ( (  Hom  `  D ) `  x ) )
2928adantr 451 . . . . . 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 2316 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
31 eqid 2316 . . . . . . . 8  |-  (comp `  D )  =  (comp `  D )
328ad2antrr 706 . . . . . . . 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 710 . . . . . . . 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 706 . . . . . . . 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 706 . . . . . . . 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 706 . . . . . . . 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 736 . . . . . . . 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 706 . . . . . . . . . . 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 5915 . . . . . . . . . 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 5917 . . . . . . . . 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 710 . . . . . . . . . 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 706 . . . . . . . . . 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 451 . . . . . . . . . . . . 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 2366 . . . . . . . . . . . 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 2383 . . . . . . . . . . 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 470 . . . . . . . . . 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 737 . . . . . . . . . 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 13636 . . . . . . . . 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 2390 . . . . . . . 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 13660 . . . . . . 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 13660 . . . . . . . . 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 5915 . . . . . . . . . 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 5917 . . . . . . . . 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 2358 . . . . . . . 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 5915 . . . . . . 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 2348 . . . . . 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 4134 . . . . 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 5951 . . . 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 5951 . . 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 3841 . 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 2316 . . 3  |-  (HomF `  C
)  =  (HomF `  C
)
6463, 42, 5, 6, 30hofval 14075 . 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 2316 . . 3  |-  (HomF `  D
)  =  (HomF `  D
)
66 hofpropd.d . . 3  |-  ( ph  ->  D  e.  Cat )
67 eqid 2316 . . 3  |-  ( Base `  D )  =  (
Base `  D )
6865, 66, 67, 7, 31hofval 14075 . 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 2358 1  |-  ( ph  ->  (HomF
`  C )  =  (HomF
`  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   <.cop 3677    e. cmpt 4114    X. cxp 4724   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   1stc1st 6162   2ndc2nd 6163   Basecbs 13195    Hom chom 13266  compcco 13267   Catccat 13615    Homf chomf 13617  compfccomf 13618  HomFchof 14071
This theorem is referenced by:  yonpropd  14091  oppcyon  14092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-cat 13619  df-homf 13621  df-comf 13622  df-hof 14073
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