Theorem hofpropd 18268

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 17685	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
3	2	sqxpeqd 5714	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
4	3	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
5		eqid 2728	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		eqid 2728	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7		eqid 2728	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
8	1	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (Homf ‘𝐶) = (Homf ‘𝐷))
9		xp1st 8033	. . . . . . 7 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑦) ∈ (Base‘𝐶))
10	9	ad2antll 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (1st ‘𝑦) ∈ (Base‘𝐶))
11		xp1st 8033	. . . . . . 7 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑥) ∈ (Base‘𝐶))
12	11	ad2antrl 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (1st ‘𝑥) ∈ (Base‘𝐶))
13	5, 6, 7, 8, 10, 12	homfeqval 17686	. . . . 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 8034	. . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑥) ∈ (Base‘𝐶))
15	14	ad2antrl 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
16		xp2nd 8034	. . . . . . . 8 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑦) ∈ (Base‘𝐶))
17	16	ad2antll 727	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (2nd ‘𝑦) ∈ (Base‘𝐶))
18	5, 6, 7, 8, 15, 17	homfeqval 17686	. . . . . 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 479	. . . . 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 17686	. . . . . . . . 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 7429	. . . . . . . . 9 ⊢ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
22		df-ov 7429	. . . . . . . . 9 ⊢ ((1st ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑥)) = ((Hom ‘𝐷)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
23	20, 21, 22	3eqtr3g 2791	. . . . . . . 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 8040	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
25	24	ad2antrl 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
26	25	fveq2d 6906	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
27	25	fveq2d 6906	. . . . . . . 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 2778	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐷)‘𝑥))
29	28	adantr 479	. . . . . 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 2728	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
31		eqid 2728	. . . . . . . 8 ⊢ (comp‘𝐷) = (comp‘𝐷)
32	8	ad2antrr 724	. . . . . . . 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 728	. . . . . . . 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 724	. . . . . . . 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 724	. . . . . . . 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 724	. . . . . . . 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 775	. . . . . . . 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 724	. . . . . . . . . . 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 7441	. . . . . . . . . 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 7443	. . . . . . . . 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 728	. . . . . . . . . 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 724	. . . . . . . . . 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 479	. . . . . . . . . . . . 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 2786	. . . . . . . . . . . 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 2815	. . . . . . . . . . 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 475	. . . . . . . . . 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 776	. . . . . . . . . 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 17674	. . . . . . . . 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 2829	. . . . . . . 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 17697	. . . . . . 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 17697	. . . . . . . . 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 7441	. . . . . . . . . 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 7443	. . . . . . . . 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 2778	. . . . . . . 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 7441	. . . . . . 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 2768	. . . . . 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 5241	. . . . 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 7501	. . . 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 7501	. . 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 4886	. 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 2728	. . 3 ⊢ (HomF‘𝐶) = (HomF‘𝐶)
64	63, 42, 5, 6, 30	hofval 18253	. 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 2728	. . 3 ⊢ (HomF‘𝐷) = (HomF‘𝐷)
66		hofpropd.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
67		eqid 2728	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
68	65, 66, 67, 7, 31	hofval 18253	. 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 2778	1 ⊢ (𝜑 → (HomF‘𝐶) = (HomF‘𝐷))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ⟨cop 4638   ↦ cmpt 5235   × cxp 5680  ‘cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17189  Hom chom 17253  compcco 17254  Catccat 17653  Hom_f chomf 17655  comp_fccomf 17656  Hom_Fchof 18249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-cat 17657  df-homf 17659  df-comf 17660  df-hof 18251

This theorem is referenced by:  yonpropd  18269  oppcyon  18270