MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hofpropd Structured version   Unicode version

Theorem hofpropd 15073
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  ->  ( Hom f  `  C )  =  ( Hom f  `  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  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
21homfeqbas 14631 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
32, 2xpeq12d 4861 . . . 4  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
43adantr 462 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) )  ->  ( ( Base `  C )  X.  ( Base `  C ) )  =  ( ( Base `  D )  X.  ( Base `  D ) ) )
5 eqid 2441 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
6 eqid 2441 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
7 eqid 2441 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
81adantr 462 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
9 xp1st 6605 . . . . . . 7  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 1st `  y
)  e.  ( Base `  C ) )
109ad2antll 723 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 1st `  y
)  e.  ( Base `  C ) )
11 xp1st 6605 . . . . . . 7  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
1211ad2antrl 722 . . . . . 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 14632 . . . . 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 6606 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 2nd `  x
)  e.  ( Base `  C ) )
1514ad2antrl 722 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  C ) )
16 xp2nd 6606 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 2nd `  y
)  e.  ( Base `  C ) )
1716ad2antll 723 . . . . . . 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 14632 . . . . . 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 462 . . . . 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 14632 . . . . . . . . 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 6093 . . . . . . . . 9  |-  ( ( 1st `  x ) ( Hom  `  C
) ( 2nd `  x
) )  =  ( ( Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
22 df-ov 6093 . . . . . . . . 9  |-  ( ( 1st `  x ) ( Hom  `  D
) ( 2nd `  x
) )  =  ( ( Hom  `  D
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
2320, 21, 223eqtr3g 2496 . . . . . . . 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 6612 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
2524ad2antrl 722 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
2625fveq2d 5692 . . . . . . . 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 5692 . . . . . . . 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 2483 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( ( Hom  `  C ) `  x
)  =  ( ( Hom  `  D ) `  x ) )
2928adantr 462 . . . . . 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 2441 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
31 eqid 2441 . . . . . . . 8  |-  (comp `  D )  =  (comp `  D )
328ad2antrr 720 . . . . . . . 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 )
)  ->  ( Hom f  `  C
)  =  ( Hom f  `  D ) )
33 hofpropd.2 . . . . . . . . 9  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
3433ad3antrrr 724 . . . . . . . 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 720 . . . . . . . 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 720 . . . . . . . 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 720 . . . . . . . 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 754 . . . . . . . 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 720 . . . . . . . . . . 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 6105 . . . . . . . . . 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 6107 . . . . . . . . 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 724 . . . . . . . . . 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 720 . . . . . . . . . 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 462 . . . . . . . . . . . . 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 2491 . . . . . . . . . . . 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 2508 . . . . . . . . . . 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 481 . . . . . . . . . 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 755 . . . . . . . . . 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 14619 . . . . . . . . 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 2515 . . . . . . . 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 14643 . . . . . . 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 14643 . . . . . . . . 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 6105 . . . . . . . . . 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 6107 . . . . . . . . 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 2483 . . . . . . . 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 6105 . . . . . . 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 2473 . . . . . 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 4366 . . . . 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 6146 . . . 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 6146 . . 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 4064 . 2  |-  ( ph  -> 
<. ( Hom f  `  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 ) ) ) ) >.  =  <. ( Hom f  `  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 2441 . . 3  |-  (HomF `  C
)  =  (HomF `  C
)
6463, 42, 5, 6, 30hofval 15058 . 2  |-  ( ph  ->  (HomF
`  C )  = 
<. ( Hom f  `  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 2441 . . 3  |-  (HomF `  D
)  =  (HomF `  D
)
66 hofpropd.d . . 3  |-  ( ph  ->  D  e.  Cat )
67 eqid 2441 . . 3  |-  ( Base `  D )  =  (
Base `  D )
6865, 66, 67, 7, 31hofval 15058 . 2  |-  ( ph  ->  (HomF
`  D )  = 
<. ( Hom f  `  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 2483 1  |-  ( ph  ->  (HomF
`  C )  =  (HomF
`  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   <.cop 3880    e. cmpt 4347    X. cxp 4834   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   Basecbs 14170   Hom chom 14245  compcco 14246   Catccat 14598   Hom f chomf 14600  compfccomf 14601  HomFchof 15054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-cat 14602  df-homf 14604  df-comf 14605  df-hof 15056
This theorem is referenced by:  yonpropd  15074  oppcyon  15075
  Copyright terms: Public domain W3C validator