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Theorem homfeqval 14657
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  ->  ( Hom f  `  C )  =  ( Hom f  `  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  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
21oveqd 6129 . 2  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X ( Hom f  `  D ) Y ) )
3 eqid 2443 . . 3  |-  ( Hom f  `  C )  =  ( Hom f  `  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 14652 . 2  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X H Y ) )
9 eqid 2443 . . 3  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
10 eqid 2443 . . 3  |-  ( Base `  D )  =  (
Base `  D )
11 homfeqval.j . . 3  |-  J  =  ( Hom  `  D
)
121homfeqbas 14656 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
134, 12syl5eq 2487 . . . 4  |-  ( ph  ->  B  =  ( Base `  D ) )
146, 13eleqtrd 2519 . . 3  |-  ( ph  ->  X  e.  ( Base `  D ) )
157, 13eleqtrd 2519 . . 3  |-  ( ph  ->  Y  e.  ( Base `  D ) )
169, 10, 11, 14, 15homfval 14652 . 2  |-  ( ph  ->  ( X ( Hom f  `  D ) Y )  =  ( X J Y ) )
172, 8, 163eqtr3d 2483 1  |-  ( ph  ->  ( X H Y )  =  ( X J Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5439  (class class class)co 6112   Basecbs 14195   Hom chom 14270   Hom f chomf 14625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-homf 14629
This theorem is referenced by:  comfeq  14666  comfeqval  14668  catpropd  14669  cidpropd  14670  monpropd  14697  funcpropd  14831  fullpropd  14851  natpropd  14907  xpcpropd  15039  curfpropd  15064  hofpropd  15098
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