Theorem setccatid 18042

Description: Lemma for setccat 18043. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypothesis

Ref	Expression
setccat.c	⊢ 𝐶 = (SetCat‘𝑈)

Assertion

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

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

Proof of Theorem setccatid

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

Step	Hyp	Ref	Expression
1		setccat.c	. . 3 ⊢ 𝐶 = (SetCat‘𝑈)
2		id 22	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ 𝑉)
3	1, 2	setcbas 18036	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 = (Base‘𝐶))
4		eqidd 2738	. 2 ⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
5		eqidd 2738	. 2 ⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶))
6	1	fvexi 6848	. . 3 ⊢ 𝐶 ∈ V
7	6	a1i 11	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
8		biid 261	. 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 6812	. . . 4 ⊢ ( I ↾ 𝑥):𝑥–1-1-onto→𝑥
10		f1of 6774	. . . 4 ⊢ (( I ↾ 𝑥):𝑥–1-1-onto→𝑥 → ( I ↾ 𝑥):𝑥⟶𝑥)
11	9, 10	mp1i 13	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → ( I ↾ 𝑥):𝑥⟶𝑥)
12		simpl 482	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ 𝑉)
13		eqid 2737	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14		simpr 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
15	1, 12, 13, 14, 14	elsetchom 18039	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → (( I ↾ 𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥) ↔ ( I ↾ 𝑥):𝑥⟶𝑥))
16	11, 15	mpbird 257	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → ( I ↾ 𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
17		simpl 482	. . . 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)
18		eqid 2737	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
19		simpr1l 1232	. . . 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)
20		simpr1r 1233	. . . 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)
21		simpr31 1265	. . . . 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))
22	1, 17, 13, 19, 20	elsetchom 18039	. . . . 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:𝑤⟶𝑥))
23	21, 22	mpbid 232	. . . 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:𝑤⟶𝑥)
24	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 ↾ 𝑥):𝑥⟶𝑥)
25	1, 17, 18, 19, 20, 20, 23, 24	setcco 18041	. . 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 ↾ 𝑥)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = (( I ↾ 𝑥) ∘ 𝑓))
26		fcoi2 6709	. . . 4 ⊢ (𝑓:𝑤⟶𝑥 → (( I ↾ 𝑥) ∘ 𝑓) = 𝑓)
27	23, 26	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 ↾ 𝑥) ∘ 𝑓) = 𝑓)
28	25, 27	eqtrd 2772	. 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 ↾ 𝑥)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
29		simpr2l 1234	. . . 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)
30		simpr32 1266	. . . . 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))
31	1, 17, 13, 20, 29	elsetchom 18039	. . . . 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:𝑥⟶𝑦))
32	30, 31	mpbid 232	. . . 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:𝑥⟶𝑦)
33	1, 17, 18, 20, 20, 29, 24, 32	setcco 18041	. . 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 ↾ 𝑥)) = (𝑔 ∘ ( I ↾ 𝑥)))
34		fcoi1 6708	. . . 4 ⊢ (𝑔:𝑥⟶𝑦 → (𝑔 ∘ ( I ↾ 𝑥)) = 𝑔)
35	32, 34	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 ↾ 𝑥)) = 𝑔)
36	33, 35	eqtrd 2772	. 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 ↾ 𝑥)) = 𝑔)
37	1, 17, 18, 19, 20, 29, 23, 32	setcco 18041	. . 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))
38		fco 6686	. . . . 5 ⊢ ((𝑔:𝑥⟶𝑦 ∧ 𝑓:𝑤⟶𝑥) → (𝑔 ∘ 𝑓):𝑤⟶𝑦)
39	32, 23, 38	syl2anc 585	. . . 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⟶𝑦)
40	1, 17, 13, 19, 29	elsetchom 18039	. . . 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):𝑤⟶𝑦))
41	39, 40	mpbird 257	. . 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))
42	37, 41	eqeltrd 2837	. 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))
43		coass 6224	. . . 4 ⊢ ((ℎ ∘ 𝑔) ∘ 𝑓) = (ℎ ∘ (𝑔 ∘ 𝑓))
44		simpr2r 1235	. . . . 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)
45		simpr33 1267	. . . . . . 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))
46	1, 17, 13, 29, 44	elsetchom 18039	. . . . . . 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:𝑦⟶𝑧))
47	45, 46	mpbid 232	. . . . . 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:𝑦⟶𝑧)
48		fco 6686	. . . . . 6 ⊢ ((ℎ:𝑦⟶𝑧 ∧ 𝑔:𝑥⟶𝑦) → (ℎ ∘ 𝑔):𝑥⟶𝑧)
49	47, 32, 48	syl2anc 585	. . . . 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):𝑥⟶𝑧)
50	1, 17, 18, 19, 20, 44, 23, 49	setcco 18041	. . . 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))
51	1, 17, 18, 19, 29, 44, 39, 47	setcco 18041	. . . 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)))
52	43, 50, 51	3eqtr4a 2798	. . 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)))
53	1, 17, 18, 20, 29, 44, 32, 47	setcco 18041	. . . 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))
54	53	oveq1d 7375	. . 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))
55	37	oveq2d 7376	. . 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)))
56	52, 54, 55	3eqtr4d 2782	. 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)))
57	3, 4, 5, 7, 8, 16, 28, 36, 42, 56	iscatd2 17638	1 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   ↦ cmpt 5167   I cid 5518   ↾ cres 5626   ∘ ccom 5628  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7360  Hom chom 17222  compcco 17223  Catccat 17621  Idccid 17622  SetCatcsetc 18033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-setc 18034

This theorem is referenced by:  setccat  18043  setcid  18044