Theorem oppccatid 17893

Description: Lemma for oppccat 17896. (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 2761	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17892	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝑂)
4	3	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) = (Base‘𝑂))
5		eqidd 2762	. . 3 ⊢ (𝐶 ∈ Cat → (Hom ‘𝑂) = (Hom ‘𝑂))
6		eqidd 2762	. . 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 2761	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
11		eqid 2761	. . . . 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 17856	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑦))
15	10, 1	oppchom 17889	. . . 4 ⊢ (𝑦(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑦)
16	14, 15	eleqtrrdi 2872	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝑂)𝑦))
17		eqid 2761	. . . . 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 17891	. . . 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 17889	. . . . . 6 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
24	22, 23	eleqtrdi 2871	. . . . 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 17858	. . . 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 2796	. . 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 17891	. . . 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 17889	. . . . . 6 ⊢ (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
31	29, 30	eleqtrdi 2871	. . . . 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 17857	. . . 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 2796	. . 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 17891	. . . . 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 17859	. . . . 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 2861	. . . 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 17889	. . . 4 ⊢ (𝑥(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑥)
38	36, 37	eleqtrrdi 2872	. . 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 17889	. . . . . . 7 ⊢ (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧)
42	40, 41	eleqtrdi 2871	. . . . . 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 17860	. . . . 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 17891	. . . . 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 17891	. . . . 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 2807	. . . 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 17891	. . . . 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 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(⟨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 7436	. . . 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 2806	. . 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 17855	. 2 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (𝑦 ∈ (Base‘𝐶) ↦ ((Id‘𝐶)‘𝑦))))
52	2, 11	cidfn 17853	. . . . 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 2772	. . 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 2145  Vcvv 3451  ⟨cop 4590   ↦ cmpt 5186   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  Hom chom 17439  compcco 17440  Catccat 17838  Idccid 17839  oppCatcoppc 17885

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-cat 17842  df-cid 17843  df-oppc 17886

This theorem is used by:  oppcid  17895  oppccat  17896