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Theorem homfeqd 17638
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 7418 . . . 4 (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
32ralrimivw 3142 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
43ralrimivw 3142 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
5 eqid 2724 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2724 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
7 eqidd 2725 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐶))
8 homfeqd.1 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
95, 6, 7, 8homfeq 17637 . 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 1533  wral 3053  cfv 6533  (class class class)co 7401  Basecbs 17143  Hom chom 17207  Homf chomf 17609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-homf 17613
This theorem is referenced by: (None)
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