Theorem oppccatid 17777

Description: Lemma for oppccat 17780. (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 2769	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17776	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝑂)
4	3	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) = (Base‘𝑂))
5		eqidd 2770	. . 3 ⊢ (𝐶 ∈ Cat → (Hom ‘𝑂) = (Hom ‘𝑂))
6		eqidd 2770	. . 3 ⊢ (𝐶 ∈ Cat → (comp‘𝑂) = (comp‘𝑂))
7	1	fvexi 6898	. . . 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 2769	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
11		eqid 2769	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
12		simpl 487	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
13		simpr 489	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
14	2, 10, 11, 12, 13	catidcl 17740	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑦))
15	10, 1	oppchom 17773	. . . 4 ⊢ (𝑦(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑦)
16	14, 15	eleqtrrdi 2880	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝑂)𝑦))
17		eqid 2769	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
18		simpr1l 1247	. . . . 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 1248	. . . . 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 17775	. . . 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 487	. . . . 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 1280	. . . . . 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 17773	. . . . . 6 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
24	22, 23	eleqtrdi 2879	. . . . 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 17742	. . . 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 2804	. . 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 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)))) → 𝑧 ∈ (Base‘𝐶))
28	2, 17, 1, 19, 19, 27	oppcco 17775	. . . 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 1281	. . . . . 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 17773	. . . . . 6 ⊢ (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
31	29, 30	eleqtrdi 2879	. . . . 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 17741	. . . 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 2804	. . 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 17775	. . . . 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 17743	. . . . 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 2869	. . . 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 17773	. . . 4 ⊢ (𝑥(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑥)
38	36, 37	eleqtrrdi 2880	. . 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 1250	. . . . . 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 1282	. . . . . . 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 17773	. . . . . . 7 ⊢ (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧)
42	40, 41	eleqtrdi 2879	. . . . . 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 17744	. . . . 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 17775	. . . . 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 17775	. . . . 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 2815	. . . 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 17775	. . . . 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 7428	. . . 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 7429	. . . 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 2814	. . 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 17739	. 2 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (𝑦 ∈ (Base‘𝐶) ↦ ((Id‘𝐶)‘𝑦))))
52	2, 11	cidfn 17737	. . . . 5 ⊢ (𝐶 ∈ Cat → (Id‘𝐶) Fn (Base‘𝐶))
53		dffn5 6942	. . . . 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 2780	. . 3 ⊢ (𝐶 ∈ Cat → ((Id‘𝑂) = (Id‘𝐶) ↔ (Id‘𝑂) = (𝑦 ∈ (Base‘𝐶) ↦ ((Id‘𝐶)‘𝑦))))
56	55	anbi2d 641	. 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)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   ↦ cmpt 5196   Fn wfn 6534  ‘cfv 6539  (class class class)co 7413  Basecbs 17271  Hom chom 17323  compcco 17324  Catccat 17722  Idccid 17723  oppCatcoppc 17769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-cnex 11158  ax-resscn 11159  ax-1cn 11160  ax-icn 11161  ax-addcl 11162  ax-addrcl 11163  ax-mulcl 11164  ax-mulrcl 11165  ax-mulcom 11166  ax-addass 11167  ax-mulass 11168  ax-distr 11169  ax-i2m1 11170  ax-1ne0 11171  ax-1rid 11172  ax-rnegex 11173  ax-rrecex 11174  ax-cnre 11175  ax-pre-lttri 11176  ax-pre-lttrn 11177  ax-pre-ltadd 11178  ax-pre-mulgt0 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11445  df-neg 11446  df-nn 12236  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12507  df-z 12594  df-dec 12714  df-sets 17226  df-slot 17244  df-ndx 17256  df-base 17272  df-hom 17336  df-cco 17337  df-cat 17726  df-cid 17727  df-oppc 17770

This theorem is referenced by:  oppcid  17779  oppccat  17780