Theorem oppccatid 17943

Description: Lemma for oppccat 17946. (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 2763	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17942	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝑂)
4	3	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) = (Base‘𝑂))
5		eqidd 2764	. . 3 ⊢ (𝐶 ∈ Cat → (Hom ‘𝑂) = (Hom ‘𝑂))
6		eqidd 2764	. . 3 ⊢ (𝐶 ∈ Cat → (comp‘𝑂) = (comp‘𝑂))
7	1	fvexi 6930	. . . 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 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
11		eqid 2763	. . . . 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 17906	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝐶)𝑦))
15	10, 1	oppchom 17939	. . . 4 ⊢ (𝑦(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑦)
16	14, 15	eleqtrrdi 2875	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑦 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑦) ∈ (𝑦(Hom ‘𝑂)𝑦))
17		eqid 2763	. . . . 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 17941	. . . 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 17939	. . . . . 6 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
24	22, 23	eleqtrdi 2874	. . . . 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 17908	. . . 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 2798	. . 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 17941	. . . 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 17939	. . . . . 6 ⊢ (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
31	29, 30	eleqtrdi 2874	. . . . 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 17907	. . . 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 2798	. . 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 17941	. . . . 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 17909	. . . . 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 2864	. . . 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 17939	. . . 4 ⊢ (𝑥(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑥)
38	36, 37	eleqtrrdi 2875	. . 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 17939	. . . . . . 7 ⊢ (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧)
42	40, 41	eleqtrdi 2874	. . . . . 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 17910	. . . . 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 17941	. . . . 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 17941	. . . . 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 2809	. . . 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 17941	. . . . 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 7466	. . . 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 7467	. . . 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 2808	. . 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 17905	. 2 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (𝑦 ∈ (Base‘𝐶) ↦ ((Id‘𝐶)‘𝑦))))
52	2, 11	cidfn 17903	. . . . 5 ⊢ (𝐶 ∈ Cat → (Id‘𝐶) Fn (Base‘𝐶))
53		dffn5 6974	. . . . 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 2774	. . 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 2147  Vcvv 3454  ⟨cop 4840   ↦ cmpt 5192   Fn wfn 6561  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  Hom chom 17483  compcco 17484  Catccat 17888  Idccid 17889  oppCatcoppc 17935

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-hom 17497  df-cco 17498  df-cat 17892  df-cid 17893  df-oppc 17936

This theorem is used by:  oppcid  17945  oppccat  17946