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

Theorem hofpropd 15735
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 15184 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
32sqxpeqd 5014 . . . 4  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
43adantr 463 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) )  ->  ( ( Base `  C )  X.  ( Base `  C ) )  =  ( ( Base `  D )  X.  ( Base `  D ) ) )
5 eqid 2454 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
6 eqid 2454 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
7 eqid 2454 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
81adantr 463 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
9 xp1st 6803 . . . . . . 7  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 1st `  y
)  e.  ( Base `  C ) )
109ad2antll 726 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 1st `  y
)  e.  ( Base `  C ) )
11 xp1st 6803 . . . . . . 7  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 1st `  x
)  e.  ( Base `  C ) )
1211ad2antrl 725 . . . . . 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 15185 . . . . 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 6804 . . . . . . . 8  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 2nd `  x
)  e.  ( Base `  C ) )
1514ad2antrl 725 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( 2nd `  x
)  e.  ( Base `  C ) )
16 xp2nd 6804 . . . . . . . 8  |-  ( y  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( 2nd `  y
)  e.  ( Base `  C ) )
1716ad2antll 726 . . . . . . 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 15185 . . . . . 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 463 . . . . 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 15185 . . . . . . . . 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 6273 . . . . . . . . 9  |-  ( ( 1st `  x ) ( Hom  `  C
) ( 2nd `  x
) )  =  ( ( Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
22 df-ov 6273 . . . . . . . . 9  |-  ( ( 1st `  x ) ( Hom  `  D
) ( 2nd `  x
) )  =  ( ( Hom  `  D
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
2320, 21, 223eqtr3g 2518 . . . . . . . 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 6810 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
2524ad2antrl 725 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
2625fveq2d 5852 . . . . . . . 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 5852 . . . . . . . 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 2505 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  /\  y  e.  ( ( Base `  C )  X.  ( Base `  C
) ) ) )  ->  ( ( Hom  `  C ) `  x
)  =  ( ( Hom  `  D ) `  x ) )
2928adantr 463 . . . . . 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 2454 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
31 eqid 2454 . . . . . . . 8  |-  (comp `  D )  =  (comp `  D )
328ad2antrr 723 . . . . . . . 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 727 . . . . . . . 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 723 . . . . . . . 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 723 . . . . . . . 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 723 . . . . . . . 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 759 . . . . . . . 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 723 . . . . . . . . . . 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 6285 . . . . . . . . . 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 6287 . . . . . . . . 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 727 . . . . . . . . . 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 723 . . . . . . . . . 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 463 . . . . . . . . . . . . 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 2513 . . . . . . . . . . . 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 2524 . . . . . . . . . . 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 482 . . . . . . . . . 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 760 . . . . . . . . . 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 15174 . . . . . . . . 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 2542 . . . . . . . 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 15196 . . . . . . 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 15196 . . . . . . . . 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 6285 . . . . . . . . . 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 6287 . . . . . . . . 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 2505 . . . . . . . 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 6285 . . . . . . 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 2495 . . . . . 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 4516 . . . . 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 6331 . . . 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 6331 . . 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 4211 . 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 2454 . . 3  |-  (HomF `  C
)  =  (HomF `  C
)
6463, 42, 5, 6, 30hofval 15720 . 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 2454 . . 3  |-  (HomF `  D
)  =  (HomF `  D
)
66 hofpropd.d . . 3  |-  ( ph  ->  D  e.  Cat )
67 eqid 2454 . . 3  |-  ( Base `  D )  =  (
Base `  D )
6865, 66, 67, 7, 31hofval 15720 . 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 2505 1  |-  ( ph  ->  (HomF
`  C )  =  (HomF
`  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   <.cop 4022    |-> cmpt 4497    X. cxp 4986   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1stc1st 6771   2ndc2nd 6772   Basecbs 14716   Hom chom 14795  compcco 14796   Catccat 15153   Hom f chomf 15155  compfccomf 15156  HomFchof 15716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-cat 15157  df-homf 15159  df-comf 15160  df-hof 15718
This theorem is referenced by:  yonpropd  15736  oppcyon  15737
  Copyright terms: Public domain W3C validator