Theorem hofcl 18276

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 2736	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2736	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
6	1, 2, 3, 4, 5	hofval 18269	. . 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 6894	. . . . . 6 ⊢ (Homf ‘𝐶) ∈ V
8		fvex 6894	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
9	8, 8	xpex 7752	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
10	9, 9	mpoex 8083	. . . . . 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 8004	. . . . 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 17	. . . 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 4861	. . 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 2774	. 2 ⊢ (𝜑 → 𝑀 = ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩)
15		eqid 2736	. . . . 5 ⊢ (𝑂 ×c 𝐶) = (𝑂 ×c 𝐶)
16		hofcl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
17	16, 3	oppcbas 17735	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝑂)
18	15, 17, 3	xpcbas 18195	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐶)) = (Base‘(𝑂 ×c 𝐶))
19		eqid 2736	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		eqid 2736	. . . 4 ⊢ (Hom ‘(𝑂 ×c 𝐶)) = (Hom ‘(𝑂 ×c 𝐶))
21		eqid 2736	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22		eqid 2736	. . . 4 ⊢ (Id‘(𝑂 ×c 𝐶)) = (Id‘(𝑂 ×c 𝐶))
23		eqid 2736	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
24		eqid 2736	. . . 4 ⊢ (comp‘(𝑂 ×c 𝐶)) = (comp‘(𝑂 ×c 𝐶))
25		eqid 2736	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
26	16	oppccat 17739	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
27	2, 26	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Cat)
28	15, 27, 2	xpccat 18207	. . . 4 ⊢ (𝜑 → (𝑂 ×c 𝐶) ∈ Cat)
29		hofcl.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
30		hofcl.d	. . . . . 6 ⊢ 𝐷 = (SetCat‘𝑈)
31	30	setccat 18103	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ∈ Cat)
32	29, 31	syl 17	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
33		eqid 2736	. . . . . . . 8 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
34	33, 3	homffn 17710	. . . . . . 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 6540	. . . . . 6 ⊢ ((Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ ((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ran (Homf ‘𝐶) ⊆ 𝑈))
38	35, 36, 37	sylanbrc 583	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈)
39	30, 29	setcbas 18096	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐷))
40	39	feq3d 6698	. . . . 5 ⊢ (𝜑 → ((Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷)))
41	38, 40	mpbid 232	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷))
42		eqid 2736	. . . . . 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 7443	. . . . . . 7 ⊢ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∈ V
44		ovex 7443	. . . . . . 7 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ∈ V
45	43, 44	mpoex 8083	. . . . . 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 8074	. . . . 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 6636	. . . . 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 258	. . . 4 ⊢ (𝜑 → (2nd ‘𝑀) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶))))
49	2	ad3antrrr 730	. . . . . . . . . . . 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 777	. . . . . . . . . . . . . 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 8025	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑦) ∈ (Base‘𝐶))
52	50, 51	syl 17	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 8025	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑥) ∈ (Base‘𝐶))
56	54, 55	syl 17	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 8026	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑦) ∈ (Base‘𝐶))
59	50, 58	syl 17	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 776	. . . . . . . . . . . 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 8032	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
63	54, 62	syl 17	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . 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 8026	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑥) ∈ (Base‘𝐶))
68	54, 67	syl 17	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
72	70, 71	eqtr4di 2789	. . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . 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 17702	. . . . . . . . . . . . 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 2835	. . . . . . . . . . . 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 17702	. . . . . . . . . . 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 8032	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
80	50, 79	syl 17	. . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . 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 7413	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
83	81, 82	eqtr4di 2789	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 2838	. . . . . . . . . 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 7110	. . . . . . . . 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 726	. . . . . . . . . 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 17709	. . . . . . . . . . . 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 6885	. . . . . . . . . . . . 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 7413	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)) = ((Homf ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
91	89, 90	eqtr4di 2789	. . . . . . . . . . . 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 2781	. . . . . . . . . . 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 726	. . . . . . . . . . . 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 7080	. . . . . . . . . . 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 2836	. . . . . . . . . 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 17709	. . . . . . . . . . . 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 6885	. . . . . . . . . . . . 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 7413	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Homf ‘𝐶)(2nd ‘𝑦)) = ((Homf ‘𝐶)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
99	97, 98	eqtr4di 2789	. . . . . . . . . . . 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 2781	. . . . . . . . . . 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 7080	. . . . . . . . . . 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 2836	. . . . . . . . . 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 18099	. . . . . . . . 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 257	. . . . . . . 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 7428	. . . . . . . 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 2838	. . . . . . 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 3188	. . . . . 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 2736	. . . . . . 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 8072	. . . . . 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 218	. . . . 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 7427	. . . . . . 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 7559	. . . . . . . 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 1452	. . . . . . 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 2791	. . . . . 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 2736	. . . . . . . 8 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
116		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
117		simprr 772	. . . . . . . 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 18197	. . . . . . 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 17732	. . . . . . . 8 ⊢ ((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) = ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥))
120	119	xpeq1i 5685	. . . . . . 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 2787	. . . . . 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 6699	. . . . 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 257	. . . 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 2736	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
125	2	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → 𝐶 ∈ Cat)
126	55	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (1st ‘𝑥) ∈ (Base‘𝐶))
127	126	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → (1st ‘𝑥) ∈ (Base‘𝐶))
128	67	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
129	128	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
130		simpr 484	. . . . . . . . . 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 17700	. . . . . . . . 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 7425	. . . . . . . 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 17701	. . . . . . . 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 2771	. . . . . . 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 5219	. . . . . 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 7413	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝐶 ∈ Cat)
138	3, 4, 124, 137, 126	catidcl 17699	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(1st ‘𝑥)) ∈ ((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑥)))
139	3, 4, 124, 137, 128	catidcl 17699	. . . . . . . 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 18273	. . . . . . 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 2785	. . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
143	142	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) = ((Homf ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
144	143, 90	eqtr4di 2789	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)))
145	33, 3, 4, 126, 128	homfval 17709	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
146	144, 145	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
147	146	reseq2d 5971	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf ‘𝐶)‘𝑥)) = ( I ↾ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))))
148		mptresid 6043	. . . . . . 7 ⊢ ( I ↾ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) = (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ 𝑓)
149	147, 148	eqtrdi 2787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf ‘𝐶)‘𝑥)) = (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ 𝑓))
150	135, 141, 149	3eqtr4d 2781	. . . . 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 7428	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(2nd ‘𝑀)𝑥) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
152	142	fveq2d 6885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ((Id‘(𝑂 ×c 𝐶))‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
153	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑂 ∈ Cat)
154		eqid 2736	. . . . . . . 8 ⊢ (Id‘𝑂) = (Id‘𝑂)
155	15, 153, 137, 17, 3, 154, 124, 22, 126, 128	xpcid 18206	. . . . . . 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 17738	. . . . . . . . . 10 ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
157	137, 156	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (Id‘𝑂) = (Id‘𝐶))
158	157	fveq1d 6883	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝑂)‘(1st ‘𝑥)) = ((Id‘𝐶)‘(1st ‘𝑥)))
159	158	opeq1d 4860	. . . . . . 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 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩)
161	151, 160	fveq12d 6888	. . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑈 ∈ 𝑉)
163	38	ffvelcdmda 7079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) ∈ 𝑈)
164	30, 23, 162, 163	setcid 18104	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐷)‘((Homf ‘𝐶)‘𝑥)) = ( I ↾ ((Homf ‘𝐶)‘𝑥)))
165	150, 161, 164	3eqtr4d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((𝑥(2nd ‘𝑀)𝑥)‘((Id‘(𝑂 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((Homf ‘𝐶)‘𝑥)))
166	2	3ad2ant1 1133	. . . . . 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 1133	. . . . . 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 1133	. . . . . 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 1207	. . . . . . 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 17	. . . . . 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 17	. . . . . 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 1208	. . . . . . 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 17	. . . . . 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 17	. . . . . 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 1209	. . . . . . 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 8025	. . . . . . 7 ⊢ (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st ‘𝑧) ∈ (Base‘𝐶))
177	175, 176	syl 17	. . . . . 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 8026	. . . . . . 7 ⊢ (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd ‘𝑧) ∈ (Base‘𝐶))
179	175, 178	syl 17	. . . . . 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 1202	. . . . . . . . 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 18197	. . . . . . . . 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 2837	. . . . . . . 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 8025	. . . . . . . 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 17	. . . . . . 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 2845	. . . . . 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 8026	. . . . . . 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 17	. . . . . 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 1203	. . . . . . . . 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 18197	. . . . . . . . 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 2837	. . . . . . . 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 8025	. . . . . . . 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 17	. . . . . . 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 17732	. . . . . . 7 ⊢ ((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)) = ((1st ‘𝑧)(Hom ‘𝐶)(1st ‘𝑦))
194	192, 193	eleqtrdi 2845	. . . . . 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 8026	. . . . . . 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 17	. . . . . 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 18275	. . . . 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 17	. . . . . . . 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 8032	. . . . . . . . 9 ⊢ (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
200	175, 199	syl 17	. . . . . . . 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 7428	. . . . . . 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 17	. . . . . . . . . . 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 4862	. . . . . . . . . 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 7428	. . . . . . . . 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 8032	. . . . . . . . . 10 ⊢ (𝑔 ∈ (((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)) × ((2nd ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑧))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
206	190, 205	syl 17	. . . . . . . . 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 8032	. . . . . . . . . 10 ⊢ (𝑓 ∈ (((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦))) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
208	182, 207	syl 17	. . . . . . . . 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 7431	. . . . . . . 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 2736	. . . . . . . . 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 18204	. . . . . . . 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 17734	. . . . . . . . 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 4860	. . . . . . . 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 2775	. . . . . . 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 6888	. . . . . 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 7413	. . . . . 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 2789	. . . . 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 6885	. . . . . . . . . 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 2789	. . . . . . . . 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 17709	. . . . . . . . 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 2771	. . . . . . . 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 6885	. . . . . . . . . 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 2789	. . . . . . . . 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 17709	. . . . . . . . 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 2771	. . . . . . . 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 4862	. . . . . . 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 6885	. . . . . . . . 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 7413	. . . . . . . . 9 ⊢ ((1st ‘𝑧)(Homf ‘𝐶)(2nd ‘𝑧)) = ((Homf ‘𝐶)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
229	227, 228	eqtr4di 2789	. . . . . . . 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 17709	. . . . . . . 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 2771	. . . . . . 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 7428	. . . . . 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 7428	. . . . . . . 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 6888	. . . . . . 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 7413	. . . . . . 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 2789	. . . . . 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 7428	. . . . . . . 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 6888	. . . . . . 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 7413	. . . . . . 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 2789	. . . . . 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 7431	. . . . 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 2781	. . . 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 17883	. . 3 ⊢ (𝜑 → (Homf ‘𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd ‘𝑀))
244		df-br 5125	. . 3 ⊢ ((Homf ‘𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd ‘𝑀) ↔ ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
245	243, 244	sylib 218	. 2 ⊢ (𝜑 → ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
246	14, 245	eqeltrd 2835	1 ⊢ (𝜑 → 𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ⊆ wss 3931  ⟨cop 4612   class class class wbr 5124   ↦ cmpt 5206   I cid 5552   × cxp 5657  ran crn 5660   ↾ cres 5661   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7991  2^nd c2nd 7992  Basecbs 17233  Hom chom 17287  compcco 17288  Catccat 17681  Idccid 17682  Hom_f chomf 17683  oppCatcoppc 17728   Func cfunc 17872  SetCatcsetc 18093   ×_c cxpc 18185  Hom_Fchof 18265

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-hom 17300  df-cco 17301  df-cat 17685  df-cid 17686  df-homf 17687  df-oppc 17729  df-func 17876  df-setc 18094  df-xpc 18189  df-hof 18267

This theorem is referenced by:  oppchofcl  18277  oppcyon  18286  yonedalem1  18289  yonedalem21  18290  yonedalem22  18295