Theorem oppccatid 17799

Description: Lemma for oppccat 17802. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypothesis

Ref	Expression
oppcbas.1	⊢ 𝑂 = (oppCat‘𝐶)

Assertion

Ref	Expression
oppccatid	⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶)))

Proof of Theorem oppccatid

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

Step	Hyp	Ref	Expression
1		oppcbas.1	. . . . 5 ⊢ 𝑂 = (oppCat‘𝐶)
2		eqid 2765	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17798	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝑂)
4	3	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) = (Base‘𝑂))
5		eqidd 2766	. . 3 ⊢ (𝐶 ∈ Cat → (Hom ‘𝑂) = (Hom ‘𝑂))
6		eqidd 2766	. . 3 ⊢ (𝐶 ∈ Cat → (comp‘𝑂) = (comp‘𝑂))
7	1	fvexi 6899	. . . 4 ⊢ 𝑂 ∈ V
8	7	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ V)
9		biid 264	. . 3 ⊢ (((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ↔ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))))
10		eqid 2765	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
11		eqid 2765	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
12		simpl 488	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
13		simpr 490	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
14	2, 10, 11, 12, 13	catidcl 17762	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑦))
15	10, 1	oppchom 17795	. . . 4 ⊢ (𝑦(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑦)
16	14, 15	eleqtrrdi 2876	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝑂)𝑦))
17		eqid 2765	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
18		simpr1l 1249	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → 𝑥 ∈ (Base‘𝐶))
19		simpr1r 1250	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → 𝑦 ∈ (Base‘𝐶))
20	2, 17, 1, 18, 19, 19	oppcco 17797	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (((Id‘𝐶)‘𝑦)(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑦)𝑓) = (𝑓(⟨𝑦, 𝑦⟩(comp‘𝐶)𝑥)((Id‘𝐶)‘𝑦)))
21		simpl 488	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → 𝐶 ∈ Cat)
22		simpr31 1282	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → 𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦))
23	10, 1	oppchom 17795	. . . . . 6 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
24	22, 23	eleqtrdi 2875	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))
25	2, 10, 11, 21, 19, 17, 18, 24	catrid 17764	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (𝑓(⟨𝑦, 𝑦⟩(comp‘𝐶)𝑥)((Id‘𝐶)‘𝑦)) = 𝑓)
26	20, 25	eqtrd 2800	. . 3 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (((Id‘𝐶)‘𝑦)(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑦)𝑓) = 𝑓)
27		simpr2l 1251	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → 𝑧 ∈ (Base‘𝐶))
28	2, 17, 1, 19, 19, 27	oppcco 17797	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (𝑔(⟨𝑦, 𝑦⟩(comp‘𝑂)𝑧)((Id‘𝐶)‘𝑦)) = (((Id‘𝐶)‘𝑦)(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑦)𝑔))
29		simpr32 1283	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))
30	10, 1	oppchom 17795	. . . . . 6 ⊢ (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
31	29, 30	eleqtrdi 2875	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑦))
32	2, 10, 11, 21, 27, 17, 19, 31	catlid 17763	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (((Id‘𝐶)‘𝑦)(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑦)𝑔) = 𝑔)
33	28, 32	eqtrd 2800	. . 3 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (𝑔(⟨𝑦, 𝑦⟩(comp‘𝑂)𝑧)((Id‘𝐶)‘𝑦)) = 𝑔)
34	2, 17, 1, 18, 19, 27	oppcco 17797	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓) = (𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔))
35	2, 10, 17, 21, 27, 19, 18, 31, 24	catcocl 17765	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔) ∈ (𝑧(Hom ‘𝐶)𝑥))
36	34, 35	eqeltrd 2865	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓) ∈ (𝑧(Hom ‘𝐶)𝑥))
37	10, 1	oppchom 17795	. . . 4 ⊢ (𝑥(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑥)
38	36, 37	eleqtrrdi 2876	. . 3 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓) ∈ (𝑥(Hom ‘𝑂)𝑧))
39		simpr2r 1252	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → 𝑤 ∈ (Base‘𝐶))
40		simpr33 1284	. . . . . . 7 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))
41	10, 1	oppchom 17795	. . . . . . 7 ⊢ (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧)
42	40, 41	eleqtrdi 2875	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → ℎ ∈ (𝑤(Hom ‘𝐶)𝑧))
43	2, 10, 17, 21, 39, 27, 19, 42, 31, 18, 24	catass 17766	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → ((𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔)(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑥)ℎ) = (𝑓(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑥)(𝑔(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑦)ℎ)))
44	2, 17, 1, 18, 27, 39	oppcco 17797	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (ℎ(⟨𝑥, 𝑧⟩(comp‘𝑂)𝑤)(𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔)) = ((𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔)(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑥)ℎ))
45	2, 17, 1, 18, 19, 39	oppcco 17797	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → ((𝑔(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑦)ℎ)(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑤)𝑓) = (𝑓(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑥)(𝑔(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑦)ℎ)))
46	43, 44, 45	3eqtr4rd 2811	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → ((𝑔(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑦)ℎ)(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝑂)𝑤)(𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔)))
47	2, 17, 1, 19, 27, 39	oppcco 17797	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (ℎ(⟨𝑦, 𝑧⟩(comp‘𝑂)𝑤)𝑔) = (𝑔(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑦)ℎ))
48	47	oveq1d 7434	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝑂)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑤)𝑓) = ((𝑔(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑦)ℎ)(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑤)𝑓))
49	34	oveq2d 7435	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → (ℎ(⟨𝑥, 𝑧⟩(comp‘𝑂)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝑂)𝑤)(𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔)))
50	46, 48, 49	3eqtr4d 2810	. . 3 ⊢ ((𝐶 ∈ Cat ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)))) → ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝑂)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝑂)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)))
51	4, 5, 6, 8, 9, 16, 26, 33, 38, 50	iscatd2 17761	. 2 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (𝑦 ∈ (Base‘𝐶) ↦ ((Id‘𝐶)‘𝑦))))
52	2, 11	cidfn 17759	. . . . 5 ⊢ (𝐶 ∈ Cat → (Id‘𝐶) Fn (Base‘𝐶))
53		dffn5 6943	. . . . 5 ⊢ ((Id‘𝐶) Fn (Base‘𝐶) ↔ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ ((Id‘𝐶)‘𝑦)))
54	52, 53	sylib 221	. . . 4 ⊢ (𝐶 ∈ Cat → (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ ((Id‘𝐶)‘𝑦)))
55	54	eqeq2d 2776	. . 3 ⊢ (𝐶 ∈ Cat → ((Id‘𝑂) = (Id‘𝐶) ↔ (Id‘𝑂) = (𝑦 ∈ (Base‘𝐶) ↦ ((Id‘𝐶)‘𝑦))))
56	55	anbi2d 642	. 2 ⊢ (𝐶 ∈ Cat → ((𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶)) ↔ (𝑂 ∈ Cat ∧ (Id‘𝑂) = (𝑦 ∈ (Base‘𝐶) ↦ ((Id‘𝐶)‘𝑦)))))
57	51, 56	mpbird 260	1 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597   ↦ cmpt 5194   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  Hom chom 17345  compcco 17346  Catccat 17744  Idccid 17745  oppCatcoppc 17791

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-sets 17248  df-slot 17266  df-ndx 17278  df-base 17294  df-hom 17358  df-cco 17359  df-cat 17748  df-cid 17749  df-oppc 17792

This theorem is used by:  oppcid  17801  oppccat  17802