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Theorem homfeqval 16180
 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 6566 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋(Homf𝐷)𝑌))
3 eqid 2610 . . 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 16175 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
9 eqid 2610 . . 3 (Homf𝐷) = (Homf𝐷)
10 eqid 2610 . . 3 (Base‘𝐷) = (Base‘𝐷)
11 homfeqval.j . . 3 𝐽 = (Hom ‘𝐷)
121homfeqbas 16179 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
134, 12syl5eq 2656 . . . 4 (𝜑𝐵 = (Base‘𝐷))
146, 13eleqtrd 2690 . . 3 (𝜑𝑋 ∈ (Base‘𝐷))
157, 13eleqtrd 2690 . . 3 (𝜑𝑌 ∈ (Base‘𝐷))
169, 10, 11, 14, 15homfval 16175 . 2 (𝜑 → (𝑋(Homf𝐷)𝑌) = (𝑋𝐽𝑌))
172, 8, 163eqtr3d 2652 1 (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Homf chomf 16150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-homf 16154 This theorem is referenced by:  comfeq  16189  comfeqval  16191  catpropd  16192  cidpropd  16193  monpropd  16220  funcpropd  16383  fullpropd  16403  natpropd  16459  xpcpropd  16671  curfpropd  16696  hofpropd  16730
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