Theorem hofcl 18339

Description: Closure of the Hom functor. Note that the codomain is the category SetCat‘𝑈 for any universe 𝑈 which contains each Hom-set. This corresponds to the assertion that 𝐶 be locally small (with respect to 𝑈). (Contributed by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
hofcl.m	⊢ 𝑀 = (HomF‘𝐶)
hofcl.o	⊢ 𝑂 = (oppCat‘𝐶)
hofcl.d	⊢ 𝐷 = (SetCat‘𝑈)
hofcl.c	⊢ (𝜑 → 𝐶 ∈ Cat)
hofcl.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
hofcl.h	⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)

Assertion

Ref	Expression
hofcl	⊢ (𝜑 → 𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))

Proof of Theorem hofcl

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

Step	Hyp	Ref	Expression
1		hofcl.m	. . . 4 ⊢ 𝑀 = (HomF‘𝐶)
2		hofcl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		eqid 2765	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2765	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2765	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
6	1, 2, 3, 4, 5	hofval 18332	. . 3 ⊢ (𝜑 → 𝑀 = ⟨(Homf ‘𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))⟩)
7		fvex 6898	. . . . . 6 ⊢ (Homf ‘𝐶) ∈ V
8		fvex 6898	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
9	8, 8	xpex 7758	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
10	9, 9	mpoex 8082	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)))) ∈ V
11	7, 10	op2ndd 8003	. . . . 5 ⊢ (𝑀 = ⟨(Homf ‘𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))⟩ → (2nd ‘𝑀) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)))))
12	6, 11	syl 18	. . . 4 ⊢ (𝜑 → (2nd ‘𝑀) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)))))
13	12	opeq2d 4847	. . 3 ⊢ (𝜑 → ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩ = ⟨(Homf ‘𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))⟩)
14	6, 13	eqtr4d 2803	. 2 ⊢ (𝜑 → 𝑀 = ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩)
15		eqid 2765	. . . . 5 ⊢ (𝑂 ×c 𝐶) = (𝑂 ×c 𝐶)
16		hofcl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
17	16, 3	oppcbas 17798	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝑂)
18	15, 17, 3	xpcbas 18258	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐶)) = (Base‘(𝑂 ×c 𝐶))
19		eqid 2765	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		eqid 2765	. . . 4 ⊢ (Hom ‘(𝑂 ×c 𝐶)) = (Hom ‘(𝑂 ×c 𝐶))
21		eqid 2765	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22		eqid 2765	. . . 4 ⊢ (Id‘(𝑂 ×c 𝐶)) = (Id‘(𝑂 ×c 𝐶))
23		eqid 2765	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
24		eqid 2765	. . . 4 ⊢ (comp‘(𝑂 ×c 𝐶)) = (comp‘(𝑂 ×c 𝐶))
25		eqid 2765	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
26	16	oppccat 17802	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
27	2, 26	syl 18	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Cat)
28	15, 27, 2	xpccat 18270	. . . 4 ⊢ (𝜑 → (𝑂 ×c 𝐶) ∈ Cat)
29		hofcl.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
30		hofcl.d	. . . . . 6 ⊢ 𝐷 = (SetCat‘𝑈)
31	30	setccat 18166	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ∈ Cat)
32	29, 31	syl 18	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
33		eqid 2765	. . . . . . . 8 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
34	33, 3	homffn 17773	. . . . . . 7 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
35	34	a1i 11	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
36		hofcl.h	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
37		df-f 6544	. . . . . 6 ⊢ ((Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ ((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ran (Homf ‘𝐶) ⊆ 𝑈))
38	35, 36, 37	sylanbrc 595	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈)
39	30, 29	setcbas 18159	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐷))
40	39	feq3d 6694	. . . . 5 ⊢ (𝜑 → ((Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷)))
41	38, 40	mpbid 235	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷))
42		eqid 2765	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))
43		ovex 7452	. . . . . . 7 ⊢ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∈ V
44		ovex 7452	. . . . . . 7 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ∈ V
45	43, 44	mpoex 8082	. . . . . 6 ⊢ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))) ∈ V
46	42, 45	fnmpoi 8073	. . . . 5 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)))) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶)))
47	12	fneq1d 6632	. . . . 5 ⊢ (𝜑 → ((2nd ‘𝑀) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶))) ↔ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)))) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶)))))
48	46, 47	mpbiri 261	. . . 4 ⊢ (𝜑 → (2nd ‘𝑀) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶))))
49	2	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝐶 ∈ Cat)
50		simplrr 790	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
51		xp1st 8024	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑦) ∈ (Base‘𝐶))
52	50, 51	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (1st ‘𝑦) ∈ (Base‘𝐶))
53	52	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (1st ‘𝑦) ∈ (Base‘𝐶))
54		simplrl 789	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
55		xp1st 8024	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑥) ∈ (Base‘𝐶))
56	54, 55	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (1st ‘𝑥) ∈ (Base‘𝐶))
57	56	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (1st ‘𝑥) ∈ (Base‘𝐶))
58		xp2nd 8025	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑦) ∈ (Base‘𝐶))
59	50, 58	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (2nd ‘𝑦) ∈ (Base‘𝐶))
60	59	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (2nd ‘𝑦) ∈ (Base‘𝐶))
61		simplrl 789	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)))
62		1st2nd2 8031	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
63	54, 62	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
64	63	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
65	64	oveq1d 7434	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑥(comp‘𝐶)(2nd ‘𝑦)) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦)))
66	65	oveqd 7436	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ) = (𝑔(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))ℎ))
67		xp2nd 8025	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑥) ∈ (Base‘𝐶))
68	54, 67	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
69	68	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (2nd ‘𝑥) ∈ (Base‘𝐶))
70	63	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
71		df-ov 7422	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
72	70, 71	eqtr4di 2818	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
73	72	eleq2d 2851	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↔ ℎ ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))))
74	73	biimpa 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → ℎ ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
75		simplrr 790	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
76	3, 4, 5, 49, 57, 69, 60, 74, 75	catcocl 17765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))ℎ) ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
77	66, 76	eqeltrd 2865	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ) ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
78	3, 4, 5, 49, 53, 57, 60, 61, 77	catcocl 17765	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓) ∈ ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦)))
79		1st2nd2 8031	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
80	50, 79	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
81	80	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Hom ‘𝐶)‘𝑦) = ((Hom ‘𝐶)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩))
82		df-ov 7422	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
83	81, 82	eqtr4di 2818	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Hom ‘𝐶)‘𝑦) = ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦)))
84	83	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → ((Hom ‘𝐶)‘𝑦) = ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦)))
85	78, 84	eleqtrrd 2868	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓) ∈ ((Hom ‘𝐶)‘𝑦))
86	85	fmpttd 7114	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)):((Hom ‘𝐶)‘𝑥)⟶((Hom ‘𝐶)‘𝑦))
87	29	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → 𝑈 ∈ 𝑉)
88	33, 3, 4, 56, 68	homfval 17772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
89	63	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Homf ‘𝐶)‘𝑥) = ((Homf ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
90		df-ov 7422	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)) = ((Homf ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
91	89, 90	eqtr4di 2818	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Homf ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)))
92	88, 91, 72	3eqtr4d 2810	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Homf ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘𝑥))
93	38	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈)
94	93, 54	ffvelcdmd 7084	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Homf ‘𝐶)‘𝑥) ∈ 𝑈)
95	92, 94	eqeltrrd 2866	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Hom ‘𝐶)‘𝑥) ∈ 𝑈)
96	33, 3, 4, 52, 59	homfval 17772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((1st ‘𝑦)(Homf ‘𝐶)(2nd ‘𝑦)) = ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦)))
97	80	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Homf ‘𝐶)‘𝑦) = ((Homf ‘𝐶)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩))
98		df-ov 7422	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Homf ‘𝐶)(2nd ‘𝑦)) = ((Homf ‘𝐶)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
99	97, 98	eqtr4di 2818	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Homf ‘𝐶)‘𝑦) = ((1st ‘𝑦)(Homf ‘𝐶)(2nd ‘𝑦)))
100	96, 99, 83	3eqtr4d 2810	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Homf ‘𝐶)‘𝑦) = ((Hom ‘𝐶)‘𝑦))
101	93, 50	ffvelcdmd 7084	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Homf ‘𝐶)‘𝑦) ∈ 𝑈)
102	100, 101	eqeltrrd 2866	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((Hom ‘𝐶)‘𝑦) ∈ 𝑈)
103	30, 87, 21, 95, 102	elsetchom 18162	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → ((ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)) ∈ (((Hom ‘𝐶)‘𝑥)(Hom ‘𝐷)((Hom ‘𝐶)‘𝑦)) ↔ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)):((Hom ‘𝐶)‘𝑥)⟶((Hom ‘𝐶)‘𝑦)))
104	86, 103	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)) ∈ (((Hom ‘𝐶)‘𝑥)(Hom ‘𝐷)((Hom ‘𝐶)‘𝑦)))
105	92, 100	oveq12d 7437	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (((Homf ‘𝐶)‘𝑥)(Hom ‘𝐷)((Homf ‘𝐶)‘𝑦)) = (((Hom ‘𝐶)‘𝑥)(Hom ‘𝐷)((Hom ‘𝐶)‘𝑦)))
106	104, 105	eleqtrrd 2868	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∧ 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))) → (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)) ∈ (((Homf ‘𝐶)‘𝑥)(Hom ‘𝐷)((Homf ‘𝐶)‘𝑦)))
107	106	ralrimivva 3210	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ∀𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥))∀𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))(ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)) ∈ (((Homf ‘𝐶)‘𝑥)(Hom ‘𝐷)((Homf ‘𝐶)‘𝑦)))
108		eqid 2765	. . . . . . 7 ⊢ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))) = (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)))
109	108	fmpo 8071	. . . . . 6 ⊢ (∀𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥))∀𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))(ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)) ∈ (((Homf ‘𝐶)‘𝑥)(Hom ‘𝐷)((Homf ‘𝐶)‘𝑦)) ↔ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))):(((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))⟶(((Homf ‘𝐶)‘𝑥)(Hom ‘𝐷)((Homf ‘𝐶)‘𝑦)))
110	107, 109	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))):(((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))⟶(((Homf ‘𝐶)‘𝑥)(Hom ‘𝐷)((Homf ‘𝐶)‘𝑦)))
111	12	oveqd 7436	. . . . . . 7 ⊢ (𝜑 → (𝑥(2nd ‘𝑀)𝑦) = (𝑥(𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))𝑦))
112	42	ovmpt4g 7566	. . . . . . . 8 ⊢ ((𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))) ∈ V) → (𝑥(𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))𝑦) = (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))
113	45, 112	mp3an3 1479	. . . . . . 7 ⊢ ((𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))𝑦) = (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))
114	111, 113	sylan9eq 2820	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑥(2nd ‘𝑀)𝑦) = (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))
115		eqid 2765	. . . . . . . 8 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
116		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
117		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
118	15, 18, 115, 4, 20, 116, 117	xpchom 18260	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) = (((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))))
119	4, 16	oppchom 17795	. . . . . . . 8 ⊢ ((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) = ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥))
120	119	xpeq1i 5689	. . . . . . 7 ⊢ (((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))) = (((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
121	118, 120	eqtrdi 2816	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) = (((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))))
122	114, 121	feq12d 6697	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((𝑥(2nd ‘𝑀)𝑦):(𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦)⟶(((Homf ‘𝐶)‘𝑥)(Hom ‘𝐷)((Homf ‘𝐶)‘𝑦)) ↔ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))):(((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))⟶(((Homf ‘𝐶)‘𝑥)(Hom ‘𝐷)((Homf ‘𝐶)‘𝑦))))
123	110, 122	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑥(2nd ‘𝑀)𝑦):(𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦)⟶(((Homf ‘𝐶)‘𝑥)(Hom ‘𝐷)((Homf ‘𝐶)‘𝑦)))
124		eqid 2765	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
125	2	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → 𝐶 ∈ Cat)
126	55	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (1st ‘𝑥) ∈ (Base‘𝐶))
127	126	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → (1st ‘𝑥) ∈ (Base‘𝐶))
128	67	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
129	128	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
130		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
131	3, 4, 124, 125, 127, 5, 129, 130	catlid 17763	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → (((Id‘𝐶)‘(2nd ‘𝑥))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))𝑓) = 𝑓)
132	131	oveq1d 7434	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → ((((Id‘𝐶)‘(2nd ‘𝑥))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))𝑓)(⟨(1st ‘𝑥), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))((Id‘𝐶)‘(1st ‘𝑥))) = (𝑓(⟨(1st ‘𝑥), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))((Id‘𝐶)‘(1st ‘𝑥))))
133	3, 4, 124, 125, 127, 5, 129, 130	catrid 17764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → (𝑓(⟨(1st ‘𝑥), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))((Id‘𝐶)‘(1st ‘𝑥))) = 𝑓)
134	132, 133	eqtrd 2800	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → ((((Id‘𝐶)‘(2nd ‘𝑥))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))𝑓)(⟨(1st ‘𝑥), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))((Id‘𝐶)‘(1st ‘𝑥))) = 𝑓)
135	134	mpteq2dva 5206	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ ((((Id‘𝐶)‘(2nd ‘𝑥))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))𝑓)(⟨(1st ‘𝑥), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))((Id‘𝐶)‘(1st ‘𝑥)))) = (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ 𝑓))
136		df-ov 7422	. . . . . . 7 ⊢ (((Id‘𝐶)‘(1st ‘𝑥))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)((Id‘𝐶)‘(2nd ‘𝑥))) = ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)‘⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩)
137	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝐶 ∈ Cat)
138	3, 4, 124, 137, 126	catidcl 17762	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(1st ‘𝑥)) ∈ ((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑥)))
139	3, 4, 124, 137, 128	catidcl 17762	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(2nd ‘𝑥)) ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
140	1, 137, 3, 4, 126, 128, 126, 128, 5, 138, 139	hof2val 18336	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (((Id‘𝐶)‘(1st ‘𝑥))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)((Id‘𝐶)‘(2nd ‘𝑥))) = (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ ((((Id‘𝐶)‘(2nd ‘𝑥))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))𝑓)(⟨(1st ‘𝑥), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))((Id‘𝐶)‘(1st ‘𝑥)))))
141	136, 140	eqtr3id 2814	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)‘⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩) = (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ ((((Id‘𝐶)‘(2nd ‘𝑥))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))𝑓)(⟨(1st ‘𝑥), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑥))((Id‘𝐶)‘(1st ‘𝑥)))))
142	62	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
143	142	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) = ((Homf ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
144	143, 90	eqtr4di 2818	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)))
145	33, 3, 4, 126, 128	homfval 17772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
146	144, 145	eqtrd 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
147	146	reseq2d 5980	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf ‘𝐶)‘𝑥)) = ( I ↾ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))))
148		mptresid 6055	. . . . . . 7 ⊢ ( I ↾ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) = (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ 𝑓)
149	147, 148	eqtrdi 2816	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf ‘𝐶)‘𝑥)) = (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ 𝑓))
150	135, 141, 149	3eqtr4d 2810	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)‘⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩) = ( I ↾ ((Homf ‘𝐶)‘𝑥)))
151	142, 142	oveq12d 7437	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(2nd ‘𝑀)𝑥) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
152	142	fveq2d 6889	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ((Id‘(𝑂 ×c 𝐶))‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
153	27	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑂 ∈ Cat)
154		eqid 2765	. . . . . . . 8 ⊢ (Id‘𝑂) = (Id‘𝑂)
155	15, 153, 137, 17, 3, 154, 124, 22, 126, 128	xpcid 18269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) = ⟨((Id‘𝑂)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩)
156	16, 124	oppcid 17801	. . . . . . . . . 10 ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
157	137, 156	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (Id‘𝑂) = (Id‘𝐶))
158	157	fveq1d 6887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝑂)‘(1st ‘𝑥)) = ((Id‘𝐶)‘(1st ‘𝑥)))
159	158	opeq1d 4846	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ⟨((Id‘𝑂)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩ = ⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩)
160	152, 155, 159	3eqtrd 2804	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩)
161	151, 160	fveq12d 6892	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((𝑥(2nd ‘𝑀)𝑥)‘((Id‘(𝑂 ×c 𝐶))‘𝑥)) = ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)‘⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩))
162	29	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑈 ∈ 𝑉)
163	38	ffvelcdmda 7083	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) ∈ 𝑈)
164	30, 23, 162, 163	setcid 18167	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐷)‘((Homf ‘𝐶)‘𝑥)) = ( I ↾ ((Homf ‘𝐶)‘𝑥)))
165	150, 161, 164	3eqtr4d 2810	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((𝑥(2nd ‘𝑀)𝑥)‘((Id‘(𝑂 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((Homf ‘𝐶)‘𝑥)))
166	2	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝐶 ∈ Cat)
167	29	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑈 ∈ 𝑉)
168	36	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ran (Homf ‘𝐶) ⊆ 𝑈)
169		simp21 1225	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
170	169, 55	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (1st ‘𝑥) ∈ (Base‘𝐶))
171	169, 67	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
172		simp22 1226	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
173	172, 51	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (1st ‘𝑦) ∈ (Base‘𝐶))
174	172, 58	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (2nd ‘𝑦) ∈ (Base‘𝐶))
175		simp23 1227	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)))
176		xp1st 8024	. . . . . . 7 ⊢ (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑧) ∈ (Base‘𝐶))
177	175, 176	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (1st ‘𝑧) ∈ (Base‘𝐶))
178		xp2nd 8025	. . . . . . 7 ⊢ (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑧) ∈ (Base‘𝐶))
179	175, 178	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (2nd ‘𝑧) ∈ (Base‘𝐶))
180		simp3l 1220	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦))
181	15, 18, 115, 4, 20, 169, 172	xpchom 18260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) = (((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))))
182	180, 181	eleqtrd 2867	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑓 ∈ (((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))))
183		xp1st 8024	. . . . . . . 8 ⊢ (𝑓 ∈ (((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))) → (1st ‘𝑓) ∈ ((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)))
184	182, 183	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (1st ‘𝑓) ∈ ((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)))
185	184, 119	eleqtrdi 2875	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (1st ‘𝑓) ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)))
186		xp2nd 8025	. . . . . . 7 ⊢ (𝑓 ∈ (((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))) → (2nd ‘𝑓) ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
187	182, 186	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (2nd ‘𝑓) ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
188		simp3r 1221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))
189	15, 18, 115, 4, 20, 172, 175	xpchom 18260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧) = (((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))))
190	188, 189	eleqtrd 2867	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑔 ∈ (((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))))
191		xp1st 8024	. . . . . . . 8 ⊢ (𝑔 ∈ (((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))) → (1st ‘𝑔) ∈ ((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)))
192	190, 191	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (1st ‘𝑔) ∈ ((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)))
193	4, 16	oppchom 17795	. . . . . . 7 ⊢ ((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)) = ((1st ‘𝑧)(Hom ‘𝐶)(1st ‘𝑦))
194	192, 193	eleqtrdi 2875	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (1st ‘𝑔) ∈ ((1st ‘𝑧)(Hom ‘𝐶)(1st ‘𝑦)))
195		xp2nd 8025	. . . . . . 7 ⊢ (𝑔 ∈ (((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))) → (2nd ‘𝑔) ∈ ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧)))
196	190, 195	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (2nd ‘𝑔) ∈ ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧)))
197	1, 16, 30, 166, 167, 168, 3, 4, 170, 171, 173, 174, 177, 179, 185, 187, 194, 196	hofcllem 18338	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (((1st ‘𝑓)(⟨(1st ‘𝑧), (1st ‘𝑦)⟩(comp‘𝐶)(1st ‘𝑥))(1st ‘𝑔))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐶)(2nd ‘𝑧))(2nd ‘𝑓))) = (((1st ‘𝑔)(⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)(2nd ‘𝑔))(⟨((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)), ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑧)(Hom ‘𝐶)(2nd ‘𝑧)))((1st ‘𝑓)(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)(2nd ‘𝑓))))
198	169, 62	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
199		1st2nd2 8031	. . . . . . . . 9 ⊢ (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
200	175, 199	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
201	198, 200	oveq12d 7437	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (𝑥(2nd ‘𝑀)𝑧) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
202	172, 79	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
203	198, 202	opeq12d 4848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ⟨𝑥, 𝑦⟩ = ⟨⟨(1st ‘𝑥), (2nd ‘𝑥)⟩, ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩⟩)
204	203, 200	oveq12d 7437	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (⟨𝑥, 𝑦⟩(comp‘(𝑂 ×c 𝐶))𝑧) = (⟨⟨(1st ‘𝑥), (2nd ‘𝑥)⟩, ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩⟩(comp‘(𝑂 ×c 𝐶))⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
205		1st2nd2 8031	. . . . . . . . . 10 ⊢ (𝑔 ∈ (((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
206	190, 205	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
207		1st2nd2 8031	. . . . . . . . . 10 ⊢ (𝑓 ∈ (((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
208	182, 207	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
209	204, 206, 208	oveq123d 7440	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑂 ×c 𝐶))𝑧)𝑓) = (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩(⟨⟨(1st ‘𝑥), (2nd ‘𝑥)⟩, ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩⟩(comp‘(𝑂 ×c 𝐶))⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)⟨(1st ‘𝑓), (2nd ‘𝑓)⟩))
210		eqid 2765	. . . . . . . . 9 ⊢ (comp‘𝑂) = (comp‘𝑂)
211	15, 17, 3, 115, 4, 170, 171, 173, 174, 210, 5, 24, 177, 179, 184, 187, 192, 196	xpcco2 18267	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩(⟨⟨(1st ‘𝑥), (2nd ‘𝑥)⟩, ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩⟩(comp‘(𝑂 ×c 𝐶))⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)⟨(1st ‘𝑓), (2nd ‘𝑓)⟩) = ⟨((1st ‘𝑔)(⟨(1st ‘𝑥), (1st ‘𝑦)⟩(comp‘𝑂)(1st ‘𝑧))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐶)(2nd ‘𝑧))(2nd ‘𝑓))⟩)
212	3, 5, 16, 170, 173, 177	oppcco 17797	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((1st ‘𝑔)(⟨(1st ‘𝑥), (1st ‘𝑦)⟩(comp‘𝑂)(1st ‘𝑧))(1st ‘𝑓)) = ((1st ‘𝑓)(⟨(1st ‘𝑧), (1st ‘𝑦)⟩(comp‘𝐶)(1st ‘𝑥))(1st ‘𝑔)))
213	212	opeq1d 4846	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ⟨((1st ‘𝑔)(⟨(1st ‘𝑥), (1st ‘𝑦)⟩(comp‘𝑂)(1st ‘𝑧))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐶)(2nd ‘𝑧))(2nd ‘𝑓))⟩ = ⟨((1st ‘𝑓)(⟨(1st ‘𝑧), (1st ‘𝑦)⟩(comp‘𝐶)(1st ‘𝑥))(1st ‘𝑔)), ((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐶)(2nd ‘𝑧))(2nd ‘𝑓))⟩)
214	209, 211, 213	3eqtrd 2804	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑂 ×c 𝐶))𝑧)𝑓) = ⟨((1st ‘𝑓)(⟨(1st ‘𝑧), (1st ‘𝑦)⟩(comp‘𝐶)(1st ‘𝑥))(1st ‘𝑔)), ((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐶)(2nd ‘𝑧))(2nd ‘𝑓))⟩)
215	201, 214	fveq12d 6892	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((𝑥(2nd ‘𝑀)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑂 ×c 𝐶))𝑧)𝑓)) = ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)‘⟨((1st ‘𝑓)(⟨(1st ‘𝑧), (1st ‘𝑦)⟩(comp‘𝐶)(1st ‘𝑥))(1st ‘𝑔)), ((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐶)(2nd ‘𝑧))(2nd ‘𝑓))⟩))
216		df-ov 7422	. . . . . 6 ⊢ (((1st ‘𝑓)(⟨(1st ‘𝑧), (1st ‘𝑦)⟩(comp‘𝐶)(1st ‘𝑥))(1st ‘𝑔))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐶)(2nd ‘𝑧))(2nd ‘𝑓))) = ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)‘⟨((1st ‘𝑓)(⟨(1st ‘𝑧), (1st ‘𝑦)⟩(comp‘𝐶)(1st ‘𝑥))(1st ‘𝑔)), ((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐶)(2nd ‘𝑧))(2nd ‘𝑓))⟩)
217	215, 216	eqtr4di 2818	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((𝑥(2nd ‘𝑀)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑂 ×c 𝐶))𝑧)𝑓)) = (((1st ‘𝑓)(⟨(1st ‘𝑧), (1st ‘𝑦)⟩(comp‘𝐶)(1st ‘𝑥))(1st ‘𝑔))(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐶)(2nd ‘𝑧))(2nd ‘𝑓))))
218	198	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((Homf ‘𝐶)‘𝑥) = ((Homf ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
219	218, 90	eqtr4di 2818	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((Homf ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)))
220	33, 3, 4, 170, 171	homfval 17772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
221	219, 220	eqtrd 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((Homf ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
222	202	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((Homf ‘𝐶)‘𝑦) = ((Homf ‘𝐶)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩))
223	222, 98	eqtr4di 2818	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((Homf ‘𝐶)‘𝑦) = ((1st ‘𝑦)(Homf ‘𝐶)(2nd ‘𝑦)))
224	33, 3, 4, 173, 174	homfval 17772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((1st ‘𝑦)(Homf ‘𝐶)(2nd ‘𝑦)) = ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦)))
225	223, 224	eqtrd 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((Homf ‘𝐶)‘𝑦) = ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦)))
226	221, 225	opeq12d 4848	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ⟨((Homf ‘𝐶)‘𝑥), ((Homf ‘𝐶)‘𝑦)⟩ = ⟨((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)), ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦))⟩)
227	200	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((Homf ‘𝐶)‘𝑧) = ((Homf ‘𝐶)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
228		df-ov 7422	. . . . . . . . 9 ⊢ ((1st ‘𝑧)(Homf ‘𝐶)(2nd ‘𝑧)) = ((Homf ‘𝐶)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
229	227, 228	eqtr4di 2818	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((Homf ‘𝐶)‘𝑧) = ((1st ‘𝑧)(Homf ‘𝐶)(2nd ‘𝑧)))
230	33, 3, 4, 177, 179	homfval 17772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((1st ‘𝑧)(Homf ‘𝐶)(2nd ‘𝑧)) = ((1st ‘𝑧)(Hom ‘𝐶)(2nd ‘𝑧)))
231	229, 230	eqtrd 2800	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((Homf ‘𝐶)‘𝑧) = ((1st ‘𝑧)(Hom ‘𝐶)(2nd ‘𝑧)))
232	226, 231	oveq12d 7437	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (⟨((Homf ‘𝐶)‘𝑥), ((Homf ‘𝐶)‘𝑦)⟩(comp‘𝐷)((Homf ‘𝐶)‘𝑧)) = (⟨((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)), ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑧)(Hom ‘𝐶)(2nd ‘𝑧))))
233	202, 200	oveq12d 7437	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (𝑦(2nd ‘𝑀)𝑧) = (⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
234	233, 206	fveq12d 6892	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((𝑦(2nd ‘𝑀)𝑧)‘𝑔) = ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)‘⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
235		df-ov 7422	. . . . . . 7 ⊢ ((1st ‘𝑔)(⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)(2nd ‘𝑔)) = ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)‘⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
236	234, 235	eqtr4di 2818	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((𝑦(2nd ‘𝑀)𝑧)‘𝑔) = ((1st ‘𝑔)(⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)(2nd ‘𝑔)))
237	198, 202	oveq12d 7437	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (𝑥(2nd ‘𝑀)𝑦) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩))
238	237, 208	fveq12d 6892	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((𝑥(2nd ‘𝑀)𝑦)‘𝑓) = ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)‘⟨(1st ‘𝑓), (2nd ‘𝑓)⟩))
239		df-ov 7422	. . . . . . 7 ⊢ ((1st ‘𝑓)(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)(2nd ‘𝑓)) = ((⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)‘⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
240	238, 239	eqtr4di 2818	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((𝑥(2nd ‘𝑀)𝑦)‘𝑓) = ((1st ‘𝑓)(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)(2nd ‘𝑓)))
241	232, 236, 240	oveq123d 7440	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → (((𝑦(2nd ‘𝑀)𝑧)‘𝑔)(⟨((Homf ‘𝐶)‘𝑥), ((Homf ‘𝐶)‘𝑦)⟩(comp‘𝐷)((Homf ‘𝐶)‘𝑧))((𝑥(2nd ‘𝑀)𝑦)‘𝑓)) = (((1st ‘𝑔)(⟨(1st ‘𝑦), (2nd ‘𝑦)⟩(2nd ‘𝑀)⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)(2nd ‘𝑔))(⟨((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)), ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑧)(Hom ‘𝐶)(2nd ‘𝑧)))((1st ‘𝑓)(⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)(2nd ‘𝑓))))
242	197, 217, 241	3eqtr4d 2810	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ((𝑥(2nd ‘𝑀)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑂 ×c 𝐶))𝑧)𝑓)) = (((𝑦(2nd ‘𝑀)𝑧)‘𝑔)(⟨((Homf ‘𝐶)‘𝑥), ((Homf ‘𝐶)‘𝑦)⟩(comp‘𝐷)((Homf ‘𝐶)‘𝑧))((𝑥(2nd ‘𝑀)𝑦)‘𝑓)))
243	18, 19, 20, 21, 22, 23, 24, 25, 28, 32, 41, 48, 123, 165, 242	isfuncd 17946	. . 3 ⊢ (𝜑 → (Homf ‘𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd ‘𝑀))
244		df-br 5112	. . 3 ⊢ ((Homf ‘𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd ‘𝑀) ↔ ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
245	243, 244	sylib 221	. 2 ⊢ (𝜑 → ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
246	14, 245	eqeltrd 2865	1 ⊢ (𝜑 → 𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457   ⊆ wss 3906  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194   I cid 5557   × cxp 5661  ran crn 5664   ↾ cres 5665   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991  Basecbs 17293  Hom chom 17345  compcco 17346  Catccat 17744  Idccid 17745  Hom_f chomf 17746  oppCatcoppc 17791   Func cfunc 17935  SetCatcsetc 18156   ×_c cxpc 18248  Hom_Fchof 18328

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17231  df-sets 17248  df-slot 17266  df-ndx 17278  df-base 17294  df-hom 17358  df-cco 17359  df-cat 17748  df-cid 17749  df-homf 17750  df-oppc 17792  df-func 17939  df-setc 18157  df-xpc 18252  df-hof 18330

This theorem is used by:  oppchofcl  18340  oppcyon  18349  yonedalem1  18352  yonedalem21  18353  yonedalem22  18358