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Theorem homfeqd 16336
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 6652 . . . 4 (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
32ralrimivw 2964 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
43ralrimivw 2964 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
5 eqid 2620 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2620 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
7 eqidd 2621 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐶))
8 homfeqd.1 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
95, 6, 7, 8homfeq 16335 . 2 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
104, 9mpbird 247 1 (𝜑 → (Homf𝐶) = (Homf𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wral 2909  cfv 5876  (class class class)co 6635  Basecbs 15838  Hom chom 15933  Homf chomf 16308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-homf 16312
This theorem is referenced by: (None)
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