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Theorem homfeqval 14618
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 6097 . 2  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X ( Hom f  `  D ) Y ) )
3 eqid 2433 . . 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 14613 . 2  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X H Y ) )
9 eqid 2433 . . 3  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
10 eqid 2433 . . 3  |-  ( Base `  D )  =  (
Base `  D )
11 homfeqval.j . . 3  |-  J  =  ( Hom  `  D
)
121homfeqbas 14617 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
134, 12syl5eq 2477 . . . 4  |-  ( ph  ->  B  =  ( Base `  D ) )
146, 13eleqtrd 2509 . . 3  |-  ( ph  ->  X  e.  ( Base `  D ) )
157, 13eleqtrd 2509 . . 3  |-  ( ph  ->  Y  e.  ( Base `  D ) )
169, 10, 11, 14, 15homfval 14613 . 2  |-  ( ph  ->  ( X ( Hom f  `  D ) Y )  =  ( X J Y ) )
172, 8, 163eqtr3d 2473 1  |-  ( ph  ->  ( X H Y )  =  ( X J Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Basecbs 14156   Hom chom 14231   Hom f chomf 14586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-homf 14590
This theorem is referenced by:  comfeq  14627  comfeqval  14629  catpropd  14630  cidpropd  14631  monpropd  14658  funcpropd  14792  fullpropd  14812  natpropd  14868  xpcpropd  15000  curfpropd  15025  hofpropd  15059
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