MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homfeqd Structured version   Visualization version   GIF version

Theorem homfeqd 16960
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 7156 . . . 4 (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
32ralrimivw 3153 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
43ralrimivw 3153 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
5 eqid 2801 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2801 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
7 eqidd 2802 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐶))
8 homfeqd.1 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
95, 6, 7, 8homfeq 16959 . 2 (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
104, 9mpbird 260 1 (𝜑 → (Homf𝐶) = (Homf𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wral 3109  cfv 6328  (class class class)co 7139  Basecbs 16478  Hom chom 16571  Homf chomf 16932
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-homf 16936
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator