Theorem hofcl 17503

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 2821	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2821	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2821	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
6	1, 2, 3, 4, 5	hofval 17496	. . 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 6678	. . . . . 6 ⊢ (Homf ‘𝐶) ∈ V
8		fvex 6678	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
9	8, 8	xpex 7470	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
10	9, 9	mpoex 7771	. . . . . 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 7694	. . . . 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 4804	. . 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 2859	. 2 ⊢ (𝜑 → 𝑀 = ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩)
15		eqid 2821	. . . . 5 ⊢ (𝑂 ×c 𝐶) = (𝑂 ×c 𝐶)
16		hofcl.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
17	16, 3	oppcbas 16982	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝑂)
18	15, 17, 3	xpcbas 17422	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐶)) = (Base‘(𝑂 ×c 𝐶))
19		eqid 2821	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		eqid 2821	. . . 4 ⊢ (Hom ‘(𝑂 ×c 𝐶)) = (Hom ‘(𝑂 ×c 𝐶))
21		eqid 2821	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
22		eqid 2821	. . . 4 ⊢ (Id‘(𝑂 ×c 𝐶)) = (Id‘(𝑂 ×c 𝐶))
23		eqid 2821	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
24		eqid 2821	. . . 4 ⊢ (comp‘(𝑂 ×c 𝐶)) = (comp‘(𝑂 ×c 𝐶))
25		eqid 2821	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
26	16	oppccat 16986	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
27	2, 26	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Cat)
28	15, 27, 2	xpccat 17434	. . . 4 ⊢ (𝜑 → (𝑂 ×c 𝐶) ∈ Cat)
29		hofcl.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
30		hofcl.d	. . . . . 6 ⊢ 𝐷 = (SetCat‘𝑈)
31	30	setccat 17339	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ∈ Cat)
32	29, 31	syl 17	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
33		eqid 2821	. . . . . . . 8 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
34	33, 3	homffn 16957	. . . . . . 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 6354	. . . . . 6 ⊢ ((Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ ((Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ran (Homf ‘𝐶) ⊆ 𝑈))
38	35, 36, 37	sylanbrc 585	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈)
39	30, 29	setcbas 17332	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐷))
40	39	feq3d 6496	. . . . 5 ⊢ (𝜑 → ((Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷)))
41	38, 40	mpbid 234	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷))
42		eqid 2821	. . . . . 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 7183	. . . . . . 7 ⊢ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)) ∈ V
44		ovex 7183	. . . . . . 7 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ∈ V
45	43, 44	mpoex 7771	. . . . . 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 7762	. . . . 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 6441	. . . . 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 260	. . . 4 ⊢ (𝜑 → (2nd ‘𝑀) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶))))
49	2	ad3antrrr 728	. . . . . . . . . . . 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 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)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
51		xp1st 7715	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 775	. . . . . . . . . . . . . 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 7715	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 7716	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 775	. . . . . . . . . . . 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 7722	. . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 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 7165	. . . . . . . . . . . . . 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 7167	. . . . . . . . . . . . 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 7716	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . 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 7153	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
72	70, 71	syl6eqr 2874	. . . . . . . . . . . . . . . 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 2898	. . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . 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 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)))) ∧ ℎ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)))
76	3, 4, 5, 49, 57, 69, 60, 74, 75	catcocl 16950	. . . . . . . . . . . . 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 2913	. . . . . . . . . . . 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 16950	. . . . . . . . . . 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 7722	. . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . 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 7153	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Hom ‘𝐶)(2nd ‘𝑦)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
83	81, 82	syl6eqr 2874	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 2916	. . . . . . . . . 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 6874	. . . . . . . . 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 724	. . . . . . . . . 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 16956	. . . . . . . . . . . 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 6669	. . . . . . . . . . . . 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 7153	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)) = ((Homf ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
91	89, 90	syl6eqr 2874	. . . . . . . . . . . 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 2866	. . . . . . . . . . 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 724	. . . . . . . . . . . 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	ffvelrnd 6847	. . . . . . . . . . 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 2914	. . . . . . . . . 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 16956	. . . . . . . . . . . 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 6669	. . . . . . . . . . . . 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 7153	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Homf ‘𝐶)(2nd ‘𝑦)) = ((Homf ‘𝐶)‘⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
99	97, 98	syl6eqr 2874	. . . . . . . . . . . 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 2866	. . . . . . . . . . 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	ffvelrnd 6847	. . . . . . . . . . 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 2914	. . . . . . . . . 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 17335	. . . . . . . . 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 259	. . . . . . . 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 7168	. . . . . . . 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 2916	. . . . . . 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 3191	. . . . . 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 2821	. . . . . . 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 7760	. . . . . 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 220	. . . . 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 7167	. . . . . . 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 7291	. . . . . . . 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 1446	. . . . . . 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 2876	. . . . . 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 2821	. . . . . . . 8 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
116		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
117		simprr 771	. . . . . . . 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 17424	. . . . . . 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 16979	. . . . . . . 8 ⊢ ((1st ‘𝑥)(Hom ‘𝑂)(1st ‘𝑦)) = ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥))
120	119	xpeq1i 5576	. . . . . . 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	syl6eq 2872	. . . . . 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 6497	. . . . 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 259	. . . 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 2821	. . . . . . . . . 10 ⊢ (Id‘𝐶) = (Id‘𝐶)
125	2	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → 𝐶 ∈ Cat)
126	55	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (1st ‘𝑥) ∈ (Base‘𝐶))
127	126	adantr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → (1st ‘𝑥) ∈ (Base‘𝐶))
128	67	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
129	128	adantr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) → (2nd ‘𝑥) ∈ (Base‘𝐶))
130		simpr 487	. . . . . . . . . 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 16948	. . . . . . . . 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 7165	. . . . . . . 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 16949	. . . . . . . 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 2856	. . . . . . 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 5154	. . . . . 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 7153	. . . . . . 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 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝐶 ∈ Cat)
138	3, 4, 124, 137, 126	catidcl 16947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(1st ‘𝑥)) ∈ ((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑥)))
139	3, 4, 124, 137, 128	catidcl 16947	. . . . . . . 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 17500	. . . . . . 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	syl5eqr 2870	. . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
143	142	fveq2d 6669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) = ((Homf ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
144	143, 90	syl6eqr 2874	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)))
145	33, 3, 4, 126, 128	homfval 16956	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((1st ‘𝑥)(Homf ‘𝐶)(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
146	144, 145	eqtrd 2856	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
147	146	reseq2d 5848	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf ‘𝐶)‘𝑥)) = ( I ↾ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))))
148		mptresid 5913	. . . . . . 7 ⊢ ( I ↾ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥))) = (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ 𝑓)
149	147, 148	syl6eq 2872	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf ‘𝐶)‘𝑥)) = (𝑓 ∈ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) ↦ 𝑓))
150	135, 141, 149	3eqtr4d 2866	. . . . 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 7168	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(2nd ‘𝑀)𝑥) = (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩(2nd ‘𝑀)⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
152	142	fveq2d 6669	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ((Id‘(𝑂 ×c 𝐶))‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
153	27	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑂 ∈ Cat)
154		eqid 2821	. . . . . . . 8 ⊢ (Id‘𝑂) = (Id‘𝑂)
155	15, 153, 137, 17, 3, 154, 124, 22, 126, 128	xpcid 17433	. . . . . . 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 16985	. . . . . . . . . 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 6667	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝑂)‘(1st ‘𝑥)) = ((Id‘𝐶)‘(1st ‘𝑥)))
159	158	opeq1d 4803	. . . . . . 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 2860	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐶)‘(2nd ‘𝑥))⟩)
161	151, 160	fveq12d 6672	. . . . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑈 ∈ 𝑉)
163	38	ffvelrnda 6846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf ‘𝐶)‘𝑥) ∈ 𝑈)
164	30, 23, 162, 163	setcid 17340	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐷)‘((Homf ‘𝐶)‘𝑥)) = ( I ↾ ((Homf ‘𝐶)‘𝑥)))
165	150, 161, 164	3eqtr4d 2866	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((𝑥(2nd ‘𝑀)𝑥)‘((Id‘(𝑂 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((Homf ‘𝐶)‘𝑥)))
166	2	3ad2ant1 1129	. . . . . 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 1129	. . . . . 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 1129	. . . . . 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 1202	. . . . . . 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 1203	. . . . . . 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 1204	. . . . . . 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 7715	. . . . . . 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 7716	. . . . . . 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 1197	. . . . . . . . 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 17424	. . . . . . . . 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 2915	. . . . . . . 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 7715	. . . . . . . 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 2923	. . . . . 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 7716	. . . . . . 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 1198	. . . . . . . . 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 17424	. . . . . . . . 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 2915	. . . . . . . 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 7715	. . . . . . . 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 16979	. . . . . . 7 ⊢ ((1st ‘𝑦)(Hom ‘𝑂)(1st ‘𝑧)) = ((1st ‘𝑧)(Hom ‘𝐶)(1st ‘𝑦))
194	192, 193	eleqtrdi 2923	. . . . . 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 7716	. . . . . . 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 17502	. . . . 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 7722	. . . . . . . . 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 7168	. . . . . . 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 4805	. . . . . . . . . 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 7168	. . . . . . . . 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 7722	. . . . . . . . . 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 7722	. . . . . . . . . 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 7171	. . . . . . . 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 2821	. . . . . . . . 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 17431	. . . . . . . 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 16981	. . . . . . . . 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 4803	. . . . . . . 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 2860	. . . . . . 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 6672	. . . . . 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 7153	. . . . . 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	syl6eqr 2874	. . . . 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 6669	. . . . . . . . . 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	syl6eqr 2874	. . . . . . . . 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 16956	. . . . . . . . 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 2856	. . . . . . . 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 6669	. . . . . . . . . 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	syl6eqr 2874	. . . . . . . . 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 16956	. . . . . . . . 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 2856	. . . . . . . 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 4805	. . . . . . 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 6669	. . . . . . . . 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 7153	. . . . . . . . 9 ⊢ ((1st ‘𝑧)(Homf ‘𝐶)(2nd ‘𝑧)) = ((Homf ‘𝐶)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
229	227, 228	syl6eqr 2874	. . . . . . . 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 16956	. . . . . . . 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 2856	. . . . . . 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 7168	. . . . . 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 7168	. . . . . . . 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 6672	. . . . . . 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 7153	. . . . . . 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	syl6eqr 2874	. . . . . 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 7168	. . . . . . . 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 6672	. . . . . . 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 7153	. . . . . . 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	syl6eqr 2874	. . . . . 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 7171	. . . . 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 2866	. . . 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 17129	. . 3 ⊢ (𝜑 → (Homf ‘𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd ‘𝑀))
244		df-br 5060	. . 3 ⊢ ((Homf ‘𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd ‘𝑀) ↔ ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
245	243, 244	sylib 220	. 2 ⊢ (𝜑 → ⟨(Homf ‘𝐶), (2nd ‘𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
246	14, 245	eqeltrd 2913	1 ⊢ (𝜑 → 𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3495   ⊆ wss 3936  ⟨cop 4567   class class class wbr 5059   ↦ cmpt 5139   I cid 5454   × cxp 5548  ran crn 5551   ↾ cres 5552   Fn wfn 6345  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150   ∈ cmpo 7152  1^st c1st 7681  2^nd c2nd 7682  Basecbs 16477  Hom chom 16570  compcco 16571  Catccat 16929  Idccid 16930  Hom_f chomf 16931  oppCatcoppc 16975   Func cfunc 17118  SetCatcsetc 17329   ×_c cxpc 17412  Hom_Fchof 17492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-hom 16583  df-cco 16584  df-cat 16933  df-cid 16934  df-homf 16935  df-oppc 16976  df-func 17122  df-setc 17330  df-xpc 17416  df-hof 17494

This theorem is referenced by:  oppchofcl  17504  oppcyon  17513  yonedalem1  17516  yonedalem21  17517  yonedalem22  17522