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 → (Homf ‘𝐶) = (Homf ‘𝐷))

Proof of Theorem homfeqd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		homfeqd.2	. . . . 5 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
2	1	oveqd 6993	. . . 4 ⊢ (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3	2	ralrimivw 3134	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
4	3	ralrimivw 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))
9	5, 6, 7, 8	homfeq 16822	. 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
10	4, 9	mpbird 249	1 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))

Syntax hints:   → wi 4   = wceq 1507  ∀wral 3089  ‘cfv 6188  (class class class)co 6976  Basecbs 16339  Hom chom 16432  Hom_f 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)