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Theorem homfeqval 16558
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 (𝜑 → (Homf𝐶) = (Homf𝐷))
homfeqval.x (𝜑𝑋𝐵)
homfeqval.y (𝜑𝑌𝐵)
Assertion
Ref Expression
homfeqval (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))

Proof of Theorem homfeqval
StepHypRef Expression
1 homfeqval.1 . . 3 (𝜑 → (Homf𝐶) = (Homf𝐷))
21oveqd 6830 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋(Homf𝐷)𝑌))
3 eqid 2760 . . 3 (Homf𝐶) = (Homf𝐶)
4 homfeqval.b . . 3 𝐵 = (Base‘𝐶)
5 homfeqval.h . . 3 𝐻 = (Hom ‘𝐶)
6 homfeqval.x . . 3 (𝜑𝑋𝐵)
7 homfeqval.y . . 3 (𝜑𝑌𝐵)
83, 4, 5, 6, 7homfval 16553 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
9 eqid 2760 . . 3 (Homf𝐷) = (Homf𝐷)
10 eqid 2760 . . 3 (Base‘𝐷) = (Base‘𝐷)
11 homfeqval.j . . 3 𝐽 = (Hom ‘𝐷)
121homfeqbas 16557 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
134, 12syl5eq 2806 . . . 4 (𝜑𝐵 = (Base‘𝐷))
146, 13eleqtrd 2841 . . 3 (𝜑𝑋 ∈ (Base‘𝐷))
157, 13eleqtrd 2841 . . 3 (𝜑𝑌 ∈ (Base‘𝐷))
169, 10, 11, 14, 15homfval 16553 . 2 (𝜑 → (𝑋(Homf𝐷)𝑌) = (𝑋𝐽𝑌))
172, 8, 163eqtr3d 2802 1 (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6813  Basecbs 16059  Hom chom 16154  Homf chomf 16528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-homf 16532
This theorem is referenced by:  comfeq  16567  comfeqval  16569  catpropd  16570  cidpropd  16571  monpropd  16598  funcpropd  16761  fullpropd  16781  natpropd  16837  xpcpropd  17049  curfpropd  17074  hofpropd  17108
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