Theorem catccatid 18013

Description: Lemma for catccat 18015. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
catccatid.c	⊢ 𝐶 = (CatCat‘𝑈)
catccatid.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
catccatid	⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ (idfunc‘𝑥))))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝑈   𝑥,𝑉

Proof of Theorem catccatid

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

Step	Hyp	Ref	Expression
1		catccatid.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2	1	a1i 11	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐵 = (Base‘𝐶))
3		eqidd 2730	. 2 ⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		eqidd 2730	. 2 ⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶))
5		catccatid.c	. . . 4 ⊢ 𝐶 = (CatCat‘𝑈)
6	5	fvexi 6836	. . 3 ⊢ 𝐶 ∈ V
7	6	a1i 11	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
8		biid 261	. 2 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) ↔ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))))
9		id 22	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉)
10	5, 1, 9	catcbas 18008	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → 𝐵 = (𝑈 ∩ Cat))
11		inss2 4189	. . . . . 6 ⊢ (𝑈 ∩ Cat) ⊆ Cat
12	10, 11	eqsstrdi 3980	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐵 ⊆ Cat)
13	12	sselda 3935	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Cat)
14		eqid 2729	. . . . 5 ⊢ (idfunc‘𝑥) = (idfunc‘𝑥)
15	14	idfucl 17788	. . . 4 ⊢ (𝑥 ∈ Cat → (idfunc‘𝑥) ∈ (𝑥 Func 𝑥))
16	13, 15	syl 17	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (idfunc‘𝑥) ∈ (𝑥 Func 𝑥))
17		simpl 482	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉)
18		eqid 2729	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19		simpr 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
20	5, 1, 17, 18, 19, 19	catchom 18010	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 Func 𝑥))
21	16, 20	eleqtrrd 2831	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (idfunc‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
22		simpl 482	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉)
23		eqid 2729	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
24		simpr1l 1231	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤 ∈ 𝐵)
25		simpr1r 1232	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥 ∈ 𝐵)
26		simpr31 1264	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥))
27	5, 1, 22, 18, 24, 25	catchom 18010	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 Func 𝑥))
28	26, 27	eleqtrd 2830	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤 Func 𝑥))
29	25, 16	syldan 591	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (idfunc‘𝑥) ∈ (𝑥 Func 𝑥))
30	5, 1, 22, 23, 24, 25, 25, 28, 29	catcco 18012	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((idfunc‘𝑥)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = ((idfunc‘𝑥) ∘func 𝑓))
31	28, 14	cofulid 17797	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((idfunc‘𝑥) ∘func 𝑓) = 𝑓)
32	30, 31	eqtrd 2764	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((idfunc‘𝑥)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
33		simpr2l 1233	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦 ∈ 𝐵)
34		simpr32 1265	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
35	5, 1, 22, 18, 25, 33	catchom 18010	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 Func 𝑦))
36	34, 35	eleqtrd 2830	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥 Func 𝑦))
37	5, 1, 22, 23, 25, 25, 33, 29, 36	catcco 18012	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)(idfunc‘𝑥)) = (𝑔 ∘func (idfunc‘𝑥)))
38	36, 14	cofurid 17798	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘func (idfunc‘𝑥)) = 𝑔)
39	37, 38	eqtrd 2764	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)(idfunc‘𝑥)) = 𝑔)
40	28, 36	cofucl 17795	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘func 𝑓) ∈ (𝑤 Func 𝑦))
41	5, 1, 22, 23, 24, 25, 33, 28, 36	catcco 18012	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) = (𝑔 ∘func 𝑓))
42	5, 1, 22, 18, 24, 33	catchom 18010	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑦) = (𝑤 Func 𝑦))
43	40, 41, 42	3eltr4d 2843	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦))
44		simpr33 1266	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))
45		simpr2r 1234	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝐵)
46	5, 1, 22, 18, 33, 45	catchom 18010	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 Func 𝑧))
47	44, 46	eleqtrd 2830	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑦 Func 𝑧))
48	28, 36, 47	cofuass 17796	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘func 𝑔) ∘func 𝑓) = (ℎ ∘func (𝑔 ∘func 𝑓)))
49	36, 47	cofucl 17795	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ ∘func 𝑔) ∈ (𝑥 Func 𝑧))
50	5, 1, 22, 23, 24, 25, 45, 28, 49	catcco 18012	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘func 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((ℎ ∘func 𝑔) ∘func 𝑓))
51	5, 1, 22, 23, 24, 33, 45, 40, 47	catcco 18012	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘func 𝑓)) = (ℎ ∘func (𝑔 ∘func 𝑓)))
52	48, 50, 51	3eqtr4d 2774	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘func 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘func 𝑓)))
53	5, 1, 22, 23, 25, 33, 45, 36, 47	catcco 18012	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) = (ℎ ∘func 𝑔))
54	53	oveq1d 7364	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((ℎ ∘func 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓))
55	41	oveq2d 7365	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘func 𝑓)))
56	52, 54, 55	3eqtr4d 2774	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)))
57	2, 3, 4, 7, 8, 21, 32, 39, 43, 56	iscatd2 17587	1 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ (idfunc‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∩ cin 3902  ⟨cop 4583   ↦ cmpt 5173  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  Hom chom 17172  compcco 17173  Catccat 17570  Idccid 17571   Func cfunc 17761  id_funccidfu 17762   ∘_func ccofu 17763  CatCatccatc 18005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-cat 17574  df-cid 17575  df-func 17765  df-idfu 17766  df-cofu 17767  df-catc 18006

This theorem is referenced by:  catcid  18014  catccat  18015