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Theorem homfeqd 16823
Description: If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homfeqd.1 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
homfeqd.2 (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
Assertion
Ref Expression
homfeqd (𝜑 → (Homf𝐶) = (Homf𝐷))

Proof of Theorem homfeqd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 homfeqd.2 . . . . 5 (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
21oveqd 6993 . . . 4 (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
32ralrimivw 3134 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
43ralrimivw 3134 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
5 eqid 2779 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2779 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
7 eqidd 2780 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐶))
8 homfeqd.1 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
95, 6, 7, 8homfeq 16822 . 2 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
104, 9mpbird 249 1 (𝜑 → (Homf𝐶) = (Homf𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wral 3089  cfv 6188  (class class class)co 6976  Basecbs 16339  Hom chom 16432  Homf chomf 16795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-homf 16799
This theorem is referenced by: (None)
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