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Theorem homfeqval 13924
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homfeqval.b  |-  B  =  ( Base `  C
)
homfeqval.h  |-  H  =  (  Hom  `  C
)
homfeqval.j  |-  J  =  (  Hom  `  D
)
homfeqval.1  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
homfeqval.x  |-  ( ph  ->  X  e.  B )
homfeqval.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
homfeqval  |-  ( ph  ->  ( X H Y )  =  ( X J Y ) )

Proof of Theorem homfeqval
StepHypRef Expression
1 homfeqval.1 . . 3  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
21oveqd 6099 . 2  |-  ( ph  ->  ( X (  Homf  `  C ) Y )  =  ( X (  Homf 
`  D ) Y ) )
3 eqid 2437 . . 3  |-  (  Homf  `  C )  =  (  Homf 
`  C )
4 homfeqval.b . . 3  |-  B  =  ( Base `  C
)
5 homfeqval.h . . 3  |-  H  =  (  Hom  `  C
)
6 homfeqval.x . . 3  |-  ( ph  ->  X  e.  B )
7 homfeqval.y . . 3  |-  ( ph  ->  Y  e.  B )
83, 4, 5, 6, 7homfval 13919 . 2  |-  ( ph  ->  ( X (  Homf  `  C ) Y )  =  ( X H Y ) )
9 eqid 2437 . . 3  |-  (  Homf  `  D )  =  (  Homf 
`  D )
10 eqid 2437 . . 3  |-  ( Base `  D )  =  (
Base `  D )
11 homfeqval.j . . 3  |-  J  =  (  Hom  `  D
)
121homfeqbas 13923 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
134, 12syl5eq 2481 . . . 4  |-  ( ph  ->  B  =  ( Base `  D ) )
146, 13eleqtrd 2513 . . 3  |-  ( ph  ->  X  e.  ( Base `  D ) )
157, 13eleqtrd 2513 . . 3  |-  ( ph  ->  Y  e.  ( Base `  D ) )
169, 10, 11, 14, 15homfval 13919 . 2  |-  ( ph  ->  ( X (  Homf  `  D ) Y )  =  ( X J Y ) )
172, 8, 163eqtr3d 2477 1  |-  ( ph  ->  ( X H Y )  =  ( X J Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   ` cfv 5455  (class class class)co 6082   Basecbs 13470    Hom chom 13541    Homf chomf 13892
This theorem is referenced by:  comfeq  13933  comfeqval  13935  catpropd  13936  cidpropd  13937  monpropd  13964  funcpropd  14098  fullpropd  14118  natpropd  14174  xpcpropd  14306  curfpropd  14331  hofpropd  14365
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-homf 13896
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