Theorem hofpropd 18225

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

Ref	Expression
hofpropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
hofpropd.2	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
hofpropd.c	⊢ (𝜑 → 𝐶 ∈ Cat)
hofpropd.d	⊢ (𝜑 → 𝐷 ∈ Cat)

Assertion

Ref	Expression
hofpropd	⊢ (𝜑 → (HomF‘𝐶) = (HomF‘𝐷))

Proof of Theorem hofpropd

Dummy variables 𝑓 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hofpropd.1	. . 3 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2	1	homfeqbas 17654	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
3	2	sqxpeqd 5651	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
4	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
5		eqid 2739	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2739	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7		eqid 2739	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
8	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (Homf ‘𝐶) = (Homf ‘𝐷))
9		xp1st 7964	. . . . . . 7 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑦) ∈ (Base‘𝐶))
10	9	ad2antll 735	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (1st ‘𝑦) ∈ (Base‘𝐶))
11		xp1st 7964	. . . . . . 7 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑥) ∈ (Base‘𝐶))
12	11	ad2antrl 734	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (1st ‘𝑥) ∈ (Base‘𝐶))
13	5, 6, 7, 8, 10, 12	homfeqval 17655	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) = ((1st ‘𝑦)(Hom ‘𝐷)(1st ‘𝑥)))
14		xp2nd 7965	. . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑥) ∈ (Base‘𝐶))
15	14	ad2antrl 734	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
16		xp2nd 7965	. . . . . . . 8 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑦) ∈ (Base‘𝐶))
17	16	ad2antll 735	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (2nd ‘𝑦) ∈ (Base‘𝐶))
18	5, 6, 7, 8, 15, 17	homfeqval 17655	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) = ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))
19	18	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ 𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥))) → ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) = ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))
20	5, 6, 7, 8, 12, 15	homfeqval 17655	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑥)))
21		df-ov 7360	. . . . . . . . 9 ⊢ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
22		df-ov 7360	. . . . . . . . 9 ⊢ ((1st ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑥)) = ((Hom ‘𝐷)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
23	20, 21, 22	3eqtr3g 2797	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) = ((Hom ‘𝐷)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
24		1st2nd2 7971	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
25	24	ad2antrl 734	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
26	25	fveq2d 6832	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
27	25	fveq2d 6832	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐷)‘𝑥) = ((Hom ‘𝐷)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
28	23, 26, 27	3eqtr4d 2784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐷)‘𝑥))
29	28	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐷)‘𝑥))
30		eqid 2739	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
31		eqid 2739	. . . . . . . 8 ⊢ (comp‘𝐷) = (comp‘𝐷)
32	8	ad2antrr 732	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (Homf ‘𝐶) = (Homf ‘𝐷))
33		hofpropd.2	. . . . . . . . 9 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
34	33	ad3antrrr 736	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (compf‘𝐶) = (compf‘𝐷))
35	10	ad2antrr 732	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (1st ‘𝑦) ∈ (Base‘𝐶))
36	12	ad2antrr 732	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (1st ‘𝑥) ∈ (Base‘𝐶))
37	17	ad2antrr 732	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (2nd ‘𝑦) ∈ (Base‘𝐶))
38		simplrl 782	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)))
39	25	ad2antrr 732	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
40	39	oveq1d 7372	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑥(comp‘𝐶)(2nd ‘𝑦)) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦)))
41	40	oveqd 7374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ) = (𝑔(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))ℎ))
42		hofpropd.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ Cat)
43	42	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝐶 ∈ Cat)
44	15	ad2antrr 732	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (2nd ‘𝑥) ∈ (Base‘𝐶))
45	26	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
46	45, 21	eqtr4di 2792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
47	46	eleq2d 2825	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↔ ℎ ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))))
48	47	biimpa 477	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → ℎ ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
49		simplrr 783	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
50	5, 6, 30, 43, 36, 44, 37, 48, 49	catcocl 17643	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))ℎ) ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
51	41, 50	eqeltrd 2839	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ) ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
52	5, 6, 30, 31, 32, 34, 35, 36, 37, 38, 51	comfeqval 17666	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓) = ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))𝑓))
53	5, 6, 30, 31, 32, 34, 36, 44, 37, 48, 49	comfeqval 17666	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))ℎ) = (𝑔(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))ℎ))
54	39	oveq1d 7372	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑥(comp‘𝐷)(2nd ‘𝑦)) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦)))
55	54	oveqd 7374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐷)(2nd ‘𝑦))ℎ) = (𝑔(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))ℎ))
56	53, 41, 55	3eqtr4d 2784	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ) = (𝑔(𝑥(comp‘𝐷)(2nd ‘𝑦))ℎ))
57	56	oveq1d 7372	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))𝑓) = ((𝑔(𝑥(comp‘𝐷)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))𝑓))
58	52, 57	eqtrd 2774	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓) = ((𝑔(𝑥(comp‘𝐷)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))𝑓))
59	29, 58	mpteq12dva 5159	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)) = (ℎ ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))𝑓)))
60	13, 19, 59	mpoeq123dva 7431	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))) = (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐷)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))𝑓))))
61	3, 4, 60	mpoeq123dva 7431	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)))) = (𝑥 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐷)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))𝑓)))))
62	1, 61	opeq12d 4813	. 2 ⊢ (𝜑 → ⟨(Homf ‘𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))⟩ = ⟨(Homf ‘𝐷), (𝑥 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐷)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))𝑓))))⟩)
63		eqid 2739	. . 3 ⊢ (HomF‘𝐶) = (HomF‘𝐶)
64	63, 42, 5, 6, 30	hofval 18210	. 2 ⊢ (𝜑 → (HomF‘𝐶) = ⟨(Homf ‘𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))⟩)
65		eqid 2739	. . 3 ⊢ (HomF‘𝐷) = (HomF‘𝐷)
66		hofpropd.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
67		eqid 2739	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
68	65, 66, 67, 7, 31	hofval 18210	. 2 ⊢ (𝜑 → (HomF‘𝐷) = ⟨(Homf ‘𝐷), (𝑥 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐷)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐷)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐷)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐷)(2nd ‘𝑦))𝑓))))⟩)
69	62, 64, 68	3eqtr4d 2784	1 ⊢ (𝜑 → (HomF‘𝐶) = (HomF‘𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ⟨cop 4562   ↦ cmpt 5154   × cxp 5617  ‘cfv 6486  (class class class)co 7357   ∈ cmpo 7359  1^st c1st 7930  2^nd c2nd 7931  Basecbs 17171  Hom chom 17223  compcco 17224  Catccat 17622  Hom_f chomf 17624  comp_fccomf 17625  Hom_Fchof 18206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  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 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7932  df-2nd 7933  df-cat 17626  df-homf 17628  df-comf 17629  df-hof 18208

This theorem is referenced by:  yonpropd  18226  oppcyon  18227