Theorem setccatid 18009

Description: Lemma for setccat 18010. (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 18003	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 = (Base‘𝐶))
4		eqidd 2730	. 2 ⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
5		eqidd 2730	. 2 ⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶))
6	1	fvexi 6840	. . 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 6806	. . . 4 ⊢ ( I ↾ 𝑥):𝑥–1-1-onto→𝑥
10		f1of 6768	. . . 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 2729	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
14		simpr 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
15	1, 12, 13, 14, 14	elsetchom 18006	. . 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 2729	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
19		simpr1l 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)))) → 𝑤 ∈ 𝑈)
20		simpr1r 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)))) → 𝑥 ∈ 𝑈)
21		simpr31 1264	. . . . 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 18006	. . . . 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 18008	. . 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 6703	. . . 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 2764	. 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 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)))) → 𝑦 ∈ 𝑈)
30		simpr32 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)))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
31	1, 17, 13, 20, 29	elsetchom 18006	. . . . 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 18008	. . 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 6702	. . . 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 2764	. 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 18008	. . 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 6680	. . . . 5 ⊢ ((𝑔:𝑥⟶𝑦 ∧ 𝑓:𝑤⟶𝑥) → (𝑔 ∘ 𝑓):𝑤⟶𝑦)
39	32, 23, 38	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):𝑤⟶𝑦)
40	1, 17, 13, 19, 29	elsetchom 18006	. . . 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 2828	. 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 6218	. . . 4 ⊢ ((ℎ ∘ 𝑔) ∘ 𝑓) = (ℎ ∘ (𝑔 ∘ 𝑓))
44		simpr2r 1234	. . . . 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 1266	. . . . . . 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 18006	. . . . . . 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 6680	. . . . . 6 ⊢ ((ℎ:𝑦⟶𝑧 ∧ 𝑔:𝑥⟶𝑦) → (ℎ ∘ 𝑔):𝑥⟶𝑧)
49	47, 32, 48	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):𝑥⟶𝑧)
50	1, 17, 18, 19, 20, 44, 23, 49	setcco 18008	. . . 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 18008	. . . 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 2790	. . 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 18008	. . . 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 7368	. . 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 7369	. . 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 2774	. 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 17605	1 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ⟨cop 4585   ↦ cmpt 5176   I cid 5517   ↾ cres 5625   ∘ ccom 5627  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Hom chom 17190  compcco 17191  Catccat 17588  Idccid 17589  SetCatcsetc 18000

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-dec 12610  df-uz 12754  df-fz 13429  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17139  df-hom 17203  df-cco 17204  df-cat 17592  df-cid 17593  df-setc 18001

This theorem is referenced by:  setccat  18010  setcid  18011