Theorem homfeqd 17712

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 7427	. . . 4 ⊢ (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3	2	ralrimivw 3137	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
4	3	ralrimivw 3137	. 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))
9	5, 6, 7, 8	homfeq 17711	. 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
10	4, 9	mpbird 257	1 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))

Syntax hints:   → wi 4   = wceq 1540  ∀wral 3052  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  Hom chom 17287  Hom_f chomf 17683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-homf 17687

This theorem is referenced by: (None)