Theorem homfeqd 17620

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 7375	. . . 4 ⊢ (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3	2	ralrimivw 3132	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
4	3	ralrimivw 3132	. 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 17619	. 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 1541  ∀wral 3051  ‘cfv 6492  (class class class)co 7358  Basecbs 17138  Hom chom 17190  Hom_f chomf 17591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-homf 17595

This theorem is referenced by: (None)