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Theorem homfeqd 17739
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 7449 . . . 4 (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
32ralrimivw 3149 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
43ralrimivw 3149 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
5 eqid 2736 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2736 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
7 eqidd 2737 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐶))
8 homfeqd.1 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
95, 6, 7, 8homfeq 17738 . 2 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
104, 9mpbird 257 1 (𝜑 → (Homf𝐶) = (Homf𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wral 3060  cfv 6560  (class class class)co 7432  Basecbs 17248  Hom chom 17309  Homf chomf 17710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-homf 17714
This theorem is referenced by: (None)
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