Theorem homfeqd 17601

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 7366	. . . 4 ⊢ (𝜑 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3	2	ralrimivw 3125	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
4	3	ralrimivw 3125	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
5		eqid 2729	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2729	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		eqidd 2730	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶))
8		homfeqd.1	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
9	5, 6, 7, 8	homfeq 17600	. 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 3044  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  Hom chom 17172  Hom_f chomf 17572

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-homf 17576

This theorem is referenced by: (None)