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Theorem homfeqval 17576
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 7373 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋(Homf𝐷)𝑌))
3 eqid 2736 . . 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 17571 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
9 eqid 2736 . . 3 (Homf𝐷) = (Homf𝐷)
10 eqid 2736 . . 3 (Base‘𝐷) = (Base‘𝐷)
11 homfeqval.j . . 3 𝐽 = (Hom ‘𝐷)
121homfeqbas 17575 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
134, 12eqtrid 2788 . . . 4 (𝜑𝐵 = (Base‘𝐷))
146, 13eleqtrd 2840 . . 3 (𝜑𝑋 ∈ (Base‘𝐷))
157, 13eleqtrd 2840 . . 3 (𝜑𝑌 ∈ (Base‘𝐷))
169, 10, 11, 14, 15homfval 17571 . 2 (𝜑 → (𝑋(Homf𝐷)𝑌) = (𝑋𝐽𝑌))
172, 8, 163eqtr3d 2784 1 (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6496  (class class class)co 7356  Basecbs 17082  Hom chom 17143  Homf chomf 17545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7920  df-2nd 7921  df-homf 17549
This theorem is referenced by:  comfeq  17585  comfeqval  17587  catpropd  17588  cidpropd  17589  monpropd  17619  funcpropd  17786  fullpropd  17806  natpropd  17864  xpcpropd  18096  curfpropd  18121  hofpropd  18155
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