Theorem estrccatid 17848

Description: Lemma for estrccat 17849. (Contributed by AV, 8-Mar-2020.)

Hypothesis

Ref	Expression
estrccat.c	⊢ 𝐶 = (ExtStrCat‘𝑈)

Assertion

Ref	Expression
estrccatid	⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ (Base‘𝑥)))))

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

Proof of Theorem estrccatid

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

Step	Hyp	Ref	Expression
1		estrccat.c	. . 3 ⊢ 𝐶 = (ExtStrCat‘𝑈)
2		id 22	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉)
3	1, 2	estrcbas 17841	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 = (Base‘𝐶))
4		eqidd 2739	. 2 ⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
5		eqidd 2739	. 2 ⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶))
6	1	fvexi 6788	. . 3 ⊢ 𝐶 ∈ V
7	6	a1i 11	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
8		biid 260	. 2 ⊢ (((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) ↔ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))))
9		f1oi 6754	. . . 4 ⊢ ( I ↾ (Base‘𝑥)):(Base‘𝑥)–1-1-onto→(Base‘𝑥)
10		f1of 6716	. . . 4 ⊢ (( I ↾ (Base‘𝑥)):(Base‘𝑥)–1-1-onto→(Base‘𝑥) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥))
11	9, 10	mp1i 13	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥))
12		simpl 483	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ 𝑉)
13		eqid 2738	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14		simpr 485	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
15		eqid 2738	. . . 4 ⊢ (Base‘𝑥) = (Base‘𝑥)
16	1, 12, 13, 14, 14, 15, 15	elestrchom 17844	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → (( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥) ↔ ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥)))
17	11, 16	mpbird 256	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
18		simpl 483	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉)
19		eqid 2738	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
20		simpr1l 1229	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤 ∈ 𝑈)
21		simpr1r 1230	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥 ∈ 𝑈)
22		eqid 2738	. . . 4 ⊢ (Base‘𝑤) = (Base‘𝑤)
23		simpr31 1262	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥))
24	1, 18, 13, 20, 21, 22, 15	elestrchom 17844	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓:(Base‘𝑤)⟶(Base‘𝑥)))
25	23, 24	mpbid 231	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥))
26	9, 10	mp1i 13	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥))
27	1, 18, 19, 20, 21, 21, 22, 15, 15, 25, 26	estrcco 17846	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = (( I ↾ (Base‘𝑥)) ∘ 𝑓))
28		fcoi2 6649	. . . 4 ⊢ (𝑓:(Base‘𝑤)⟶(Base‘𝑥) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
29	25, 28	syl 17	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
30	27, 29	eqtrd 2778	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
31		simpr2l 1231	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦 ∈ 𝑈)
32		eqid 2738	. . . 4 ⊢ (Base‘𝑦) = (Base‘𝑦)
33		simpr32 1263	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
34	1, 18, 13, 21, 31, 15, 32	elestrchom 17844	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔:(Base‘𝑥)⟶(Base‘𝑦)))
35	33, 34	mpbid 231	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
36	1, 18, 19, 21, 21, 31, 15, 15, 32, 26, 35	estrcco 17846	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = (𝑔 ∘ ( I ↾ (Base‘𝑥))))
37		fcoi1 6648	. . . 4 ⊢ (𝑔:(Base‘𝑥)⟶(Base‘𝑦) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
38	35, 37	syl 17	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
39	36, 38	eqtrd 2778	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
40	1, 18, 19, 20, 21, 31, 22, 15, 32, 25, 35	estrcco 17846	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) = (𝑔 ∘ 𝑓))
41		fco 6624	. . . . 5 ⊢ ((𝑔:(Base‘𝑥)⟶(Base‘𝑦) ∧ 𝑓:(Base‘𝑤)⟶(Base‘𝑥)) → (𝑔 ∘ 𝑓):(Base‘𝑤)⟶(Base‘𝑦))
42	35, 25, 41	syl2anc 584	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ 𝑓):(Base‘𝑤)⟶(Base‘𝑦))
43	1, 18, 13, 20, 31, 22, 32	elestrchom 17844	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑔 ∘ 𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦) ↔ (𝑔 ∘ 𝑓):(Base‘𝑤)⟶(Base‘𝑦)))
44	42, 43	mpbird 256	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ 𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦))
45	40, 44	eqeltrd 2839	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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))
46		coass 6169	. . . 4 ⊢ ((ℎ ∘ 𝑔) ∘ 𝑓) = (ℎ ∘ (𝑔 ∘ 𝑓))
47		simpr2r 1232	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝑈)
48		eqid 2738	. . . . 5 ⊢ (Base‘𝑧) = (Base‘𝑧)
49		simpr33 1264	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))
50	1, 18, 13, 31, 47, 32, 48	elestrchom 17844	. . . . . . 7 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ ℎ:(Base‘𝑦)⟶(Base‘𝑧)))
51	49, 50	mpbid 231	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ:(Base‘𝑦)⟶(Base‘𝑧))
52		fco 6624	. . . . . 6 ⊢ ((ℎ:(Base‘𝑦)⟶(Base‘𝑧) ∧ 𝑔:(Base‘𝑥)⟶(Base‘𝑦)) → (ℎ ∘ 𝑔):(Base‘𝑥)⟶(Base‘𝑧))
53	51, 35, 52	syl2anc 584	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ ∘ 𝑔):(Base‘𝑥)⟶(Base‘𝑧))
54	1, 18, 19, 20, 21, 47, 22, 15, 48, 25, 53	estrcco 17846	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘ 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((ℎ ∘ 𝑔) ∘ 𝑓))
55	1, 18, 19, 20, 31, 47, 22, 32, 48, 42, 51	estrcco 17846	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘ 𝑓)) = (ℎ ∘ (𝑔 ∘ 𝑓)))
56	46, 54, 55	3eqtr4a 2804	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘ 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘ 𝑓)))
57	1, 18, 19, 21, 31, 47, 15, 32, 48, 35, 51	estrcco 17846	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) = (ℎ ∘ 𝑔))
58	57	oveq1d 7290	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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 ∘ 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓))
59	40	oveq2d 7291	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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 ∘ 𝑓)))
60	56, 58, 59	3eqtr4d 2788	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) ∧ (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)))
61	3, 4, 5, 7, 8, 17, 30, 39, 45, 60	iscatd2 17390	1 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ (Base‘𝑥)))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   ↦ cmpt 5157   I cid 5488   ↾ cres 5591   ∘ ccom 5593  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374  ExtStrCatcestrc 17838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-slot 16883  df-ndx 16895  df-base 16913  df-hom 16986  df-cco 16987  df-cat 17377  df-cid 17378  df-estrc 17839

This theorem is referenced by:  estrccat  17849  estrcid  17850