Theorem homfeq 17655

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 2729	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
2		eqid 2729	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		homfeq.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
4	1, 2, 3	homffval 17651	. . . 4 ⊢ (Homf ‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦))
5		homfeq.1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
6		eqidd 2730	. . . . 5 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥𝐻𝑦))
7	5, 5, 6	mpoeq123dv 7464	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥𝐻𝑦)))
8	4, 7	eqtr4id 2783	. . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
9		eqid 2729	. . . . 5 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
10		eqid 2729	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
11		homfeq.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝐷)
12	9, 10, 11	homffval 17651	. . . 4 ⊢ (Homf ‘𝐷) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦))
13		homfeq.2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
14		eqidd 2730	. . . . 5 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥𝐽𝑦))
15	13, 13, 14	mpoeq123dv 7464	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)) = (𝑥 ∈ (Base‘𝐷), 𝑦 ∈ (Base‘𝐷) ↦ (𝑥𝐽𝑦)))
16	12, 15	eqtr4id 2783	. . 3 ⊢ (𝜑 → (Homf ‘𝐷) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦)))
17	8, 16	eqeq12d 2745	. 2 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐽𝑦))))
18		ovex 7420	. . . 4 ⊢ (𝑥𝐻𝑦) ∈ V
19	18	rgen2w 3049	. . 3 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ V
20		mpo2eqb 7521	. . 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 287	1 ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  Basecbs 17179  Hom chom 17231  Hom_f chomf 17627

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-homf 17631

This theorem is referenced by:  homfeqd  17656  fullresc  17813  resssetc  18054  resscatc  18071  funchomf  49083  initopropd  49229  termopropd  49230  termolmd  49656