Theorem homfeq 17320

Description: Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
homfeq.h	⊢ 𝐻 = (Hom ‘𝐶)
homfeq.j	⊢ 𝐽 = (Hom ‘𝐷)
homfeq.1	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
homfeq.2	⊢ (𝜑 → 𝐵 = (Base‘𝐷))

Assertion

Ref	Expression
homfeq	⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦   𝑥,𝐽,𝑦

Proof of Theorem homfeq

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
2		eqid 2738	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		homfeq.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
4	1, 2, 3	homffval 17316	. . . 4 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))
5		homfeq.1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
6		eqidd 2739	. . . . 5 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦))
7	5, 5, 6	mpoeq123dv 7328	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)))
8	4, 7	eqtr4id 2798	. . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
9		eqid 2738	. . . . 5 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
10		eqid 2738	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
11		homfeq.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝐷)
12	9, 10, 11	homffval 17316	. . . 4 ⊢ (Homf ‘𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))
13		homfeq.2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
14		eqidd 2739	. . . . 5 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦))
15	13, 13, 14	mpoeq123dv 7328	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)))
16	12, 15	eqtr4id 2798	. . 3 ⊢ (𝜑 → (Homf ‘𝐷) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)))
17	8, 16	eqeq12d 2754	. 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦))))
18		ovex 7288	. . . 4 ⊢ (𝑥𝐻𝑦) ∈ V
19	18	rgen2w 3076	. . 3 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ V
20		mpo2eqb 7384	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ V → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
21	19, 20	ax-mp 5	. 2 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))
22	17, 21	bitrdi 286	1 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  Basecbs 16840  Hom chom 16899  Hom_f chomf 17292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-homf 17296

This theorem is referenced by:  homfeqd  17321  fullresc  17482  resssetc  17723  resscatc  17740