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Theorem hofcl 18316
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 = (HomF𝐶)
hofcl.o 𝑂 = (oppCat‘𝐶)
hofcl.d 𝐷 = (SetCat‘𝑈)
hofcl.c (𝜑𝐶 ∈ Cat)
hofcl.u (𝜑𝑈𝑉)
hofcl.h (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
Assertion
Ref Expression
hofcl (𝜑𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))

Proof of Theorem hofcl
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofcl.m . . . 4 𝑀 = (HomF𝐶)
2 hofcl.c . . . 4 (𝜑𝐶 ∈ Cat)
3 eqid 2735 . . . 4 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2735 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2735 . . . 4 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5hofval 18309 . . 3 (𝜑𝑀 = ⟨(Homf𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))⟩)
7 fvex 6920 . . . . . 6 (Homf𝐶) ∈ V
8 fvex 6920 . . . . . . . 8 (Base‘𝐶) ∈ V
98, 8xpex 7772 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
109, 9mpoex 8103 . . . . . 6 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)))) ∈ V
117, 10op2ndd 8024 . . . . 5 (𝑀 = ⟨(Homf𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))⟩ → (2nd𝑀) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)))))
126, 11syl 17 . . . 4 (𝜑 → (2nd𝑀) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)))))
1312opeq2d 4885 . . 3 (𝜑 → ⟨(Homf𝐶), (2nd𝑀)⟩ = ⟨(Homf𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))⟩)
146, 13eqtr4d 2778 . 2 (𝜑𝑀 = ⟨(Homf𝐶), (2nd𝑀)⟩)
15 eqid 2735 . . . . 5 (𝑂 ×c 𝐶) = (𝑂 ×c 𝐶)
16 hofcl.o . . . . . 6 𝑂 = (oppCat‘𝐶)
1716, 3oppcbas 17764 . . . . 5 (Base‘𝐶) = (Base‘𝑂)
1815, 17, 3xpcbas 18234 . . . 4 ((Base‘𝐶) × (Base‘𝐶)) = (Base‘(𝑂 ×c 𝐶))
19 eqid 2735 . . . 4 (Base‘𝐷) = (Base‘𝐷)
20 eqid 2735 . . . 4 (Hom ‘(𝑂 ×c 𝐶)) = (Hom ‘(𝑂 ×c 𝐶))
21 eqid 2735 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
22 eqid 2735 . . . 4 (Id‘(𝑂 ×c 𝐶)) = (Id‘(𝑂 ×c 𝐶))
23 eqid 2735 . . . 4 (Id‘𝐷) = (Id‘𝐷)
24 eqid 2735 . . . 4 (comp‘(𝑂 ×c 𝐶)) = (comp‘(𝑂 ×c 𝐶))
25 eqid 2735 . . . 4 (comp‘𝐷) = (comp‘𝐷)
2616oppccat 17769 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
272, 26syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
2815, 27, 2xpccat 18246 . . . 4 (𝜑 → (𝑂 ×c 𝐶) ∈ Cat)
29 hofcl.u . . . . 5 (𝜑𝑈𝑉)
30 hofcl.d . . . . . 6 𝐷 = (SetCat‘𝑈)
3130setccat 18139 . . . . 5 (𝑈𝑉𝐷 ∈ Cat)
3229, 31syl 17 . . . 4 (𝜑𝐷 ∈ Cat)
33 eqid 2735 . . . . . . . 8 (Homf𝐶) = (Homf𝐶)
3433, 3homffn 17738 . . . . . . 7 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
3534a1i 11 . . . . . 6 (𝜑 → (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
36 hofcl.h . . . . . 6 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
37 df-f 6567 . . . . . 6 ((Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ ((Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ran (Homf𝐶) ⊆ 𝑈))
3835, 36, 37sylanbrc 583 . . . . 5 (𝜑 → (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈)
3930, 29setcbas 18132 . . . . . 6 (𝜑𝑈 = (Base‘𝐷))
4039feq3d 6724 . . . . 5 (𝜑 → ((Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷)))
4138, 40mpbid 232 . . . 4 (𝜑 → (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷))
42 eqid 2735 . . . . . 6 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))
43 ovex 7464 . . . . . . 7 ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∈ V
44 ovex 7464 . . . . . . 7 ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ∈ V
4543, 44mpoex 8103 . . . . . 6 (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))) ∈ V
4642, 45fnmpoi 8094 . . . . 5 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)))) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶)))
4712fneq1d 6662 . . . . 5 (𝜑 → ((2nd𝑀) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶))) ↔ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)))) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶)))))
4846, 47mpbiri 258 . . . 4 (𝜑 → (2nd𝑀) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶))))
492ad3antrrr 730 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝐶 ∈ Cat)
50 simplrr 778 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
51 xp1st 8045 . . . . . . . . . . . . . 14 (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st𝑦) ∈ (Base‘𝐶))
5250, 51syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → (1st𝑦) ∈ (Base‘𝐶))
5352adantr 480 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (1st𝑦) ∈ (Base‘𝐶))
54 simplrl 777 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
55 xp1st 8045 . . . . . . . . . . . . . 14 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st𝑥) ∈ (Base‘𝐶))
5654, 55syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → (1st𝑥) ∈ (Base‘𝐶))
5756adantr 480 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (1st𝑥) ∈ (Base‘𝐶))
58 xp2nd 8046 . . . . . . . . . . . . . 14 (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd𝑦) ∈ (Base‘𝐶))
5950, 58syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → (2nd𝑦) ∈ (Base‘𝐶))
6059adantr 480 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (2nd𝑦) ∈ (Base‘𝐶))
61 simplrl 777 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)))
62 1st2nd2 8052 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6354, 62syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6463adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6564oveq1d 7446 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑥(comp‘𝐶)(2nd𝑦)) = (⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑦)))
6665oveqd 7448 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd𝑦))) = (𝑔(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑦))))
67 xp2nd 8046 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd𝑥) ∈ (Base‘𝐶))
6854, 67syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → (2nd𝑥) ∈ (Base‘𝐶))
6968adantr 480 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (2nd𝑥) ∈ (Base‘𝐶))
7063fveq2d 6911 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
71 df-ov 7434 . . . . . . . . . . . . . . . . 17 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
7270, 71eqtr4di 2793 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Hom ‘𝐶)‘𝑥) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
7372eleq2d 2825 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ( ∈ ((Hom ‘𝐶)‘𝑥) ↔ ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))))
7473biimpa 476 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
75 simplrr 778 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))
763, 4, 5, 49, 57, 69, 60, 74, 75catcocl 17730 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑦))) ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑦)))
7766, 76eqeltrd 2839 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → (𝑔(𝑥(comp‘𝐶)(2nd𝑦))) ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑦)))
783, 4, 5, 49, 53, 57, 60, 61, 77catcocl 17730 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓) ∈ ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)))
79 1st2nd2 8052 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
8050, 79syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
8180fveq2d 6911 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Hom ‘𝐶)‘𝑦) = ((Hom ‘𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩))
82 df-ov 7434 . . . . . . . . . . . . 13 ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)) = ((Hom ‘𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩)
8381, 82eqtr4di 2793 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Hom ‘𝐶)‘𝑦) = ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)))
8483adantr 480 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → ((Hom ‘𝐶)‘𝑦) = ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)))
8578, 84eleqtrrd 2842 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) ∧ ∈ ((Hom ‘𝐶)‘𝑥)) → ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓) ∈ ((Hom ‘𝐶)‘𝑦))
8685fmpttd 7135 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)):((Hom ‘𝐶)‘𝑥)⟶((Hom ‘𝐶)‘𝑦))
8729ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → 𝑈𝑉)
8833, 3, 4, 56, 68homfval 17737 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
8963fveq2d 6911 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Homf𝐶)‘𝑥) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
90 df-ov 7434 . . . . . . . . . . . . 13 ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
9189, 90eqtr4di 2793 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Homf𝐶)(2nd𝑥)))
9288, 91, 723eqtr4d 2785 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Homf𝐶)‘𝑥) = ((Hom ‘𝐶)‘𝑥))
9338ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈)
9493, 54ffvelcdmd 7105 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Homf𝐶)‘𝑥) ∈ 𝑈)
9592, 94eqeltrrd 2840 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Hom ‘𝐶)‘𝑥) ∈ 𝑈)
9633, 3, 4, 52, 59homfval 17737 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((1st𝑦)(Homf𝐶)(2nd𝑦)) = ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)))
9780fveq2d 6911 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Homf𝐶)‘𝑦) = ((Homf𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩))
98 df-ov 7434 . . . . . . . . . . . . 13 ((1st𝑦)(Homf𝐶)(2nd𝑦)) = ((Homf𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩)
9997, 98eqtr4di 2793 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Homf𝐶)‘𝑦) = ((1st𝑦)(Homf𝐶)(2nd𝑦)))
10096, 99, 833eqtr4d 2785 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Homf𝐶)‘𝑦) = ((Hom ‘𝐶)‘𝑦))
10193, 50ffvelcdmd 7105 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Homf𝐶)‘𝑦) ∈ 𝑈)
102100, 101eqeltrrd 2840 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ((Hom ‘𝐶)‘𝑦) ∈ 𝑈)
10330, 87, 21, 95, 102elsetchom 18135 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → (( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)) ∈ (((Hom ‘𝐶)‘𝑥)(Hom ‘𝐷)((Hom ‘𝐶)‘𝑦)) ↔ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)):((Hom ‘𝐶)‘𝑥)⟶((Hom ‘𝐶)‘𝑦)))
10486, 103mpbird 257 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)) ∈ (((Hom ‘𝐶)‘𝑥)(Hom ‘𝐷)((Hom ‘𝐶)‘𝑦)))
10592, 100oveq12d 7449 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → (((Homf𝐶)‘𝑥)(Hom ‘𝐷)((Homf𝐶)‘𝑦)) = (((Hom ‘𝐶)‘𝑥)(Hom ‘𝐷)((Hom ‘𝐶)‘𝑦)))
106104, 105eleqtrrd 2842 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)) ∈ (((Homf𝐶)‘𝑥)(Hom ‘𝐷)((Homf𝐶)‘𝑦)))
107106ralrimivva 3200 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ∀𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥))∀𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)) ∈ (((Homf𝐶)‘𝑥)(Hom ‘𝐷)((Homf𝐶)‘𝑦)))
108 eqid 2735 . . . . . . 7 (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))) = (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)))
109108fmpo 8092 . . . . . 6 (∀𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥))∀𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)) ∈ (((Homf𝐶)‘𝑥)(Hom ‘𝐷)((Homf𝐶)‘𝑦)) ↔ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))):(((1st𝑦)(Hom ‘𝐶)(1st𝑥)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))⟶(((Homf𝐶)‘𝑥)(Hom ‘𝐷)((Homf𝐶)‘𝑦)))
110107, 109sylib 218 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))):(((1st𝑦)(Hom ‘𝐶)(1st𝑥)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))⟶(((Homf𝐶)‘𝑥)(Hom ‘𝐷)((Homf𝐶)‘𝑦)))
11112oveqd 7448 . . . . . . 7 (𝜑 → (𝑥(2nd𝑀)𝑦) = (𝑥(𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))𝑦))
11242ovmpt4g 7580 . . . . . . . 8 ((𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))) ∈ V) → (𝑥(𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))𝑦) = (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))
11345, 112mp3an3 1449 . . . . . . 7 ((𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))𝑦) = (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))
114111, 113sylan9eq 2795 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑥(2nd𝑀)𝑦) = (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))))
115 eqid 2735 . . . . . . . 8 (Hom ‘𝑂) = (Hom ‘𝑂)
116 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
117 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
11815, 18, 115, 4, 20, 116, 117xpchom 18236 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) = (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))))
1194, 16oppchom 17761 . . . . . . . 8 ((1st𝑥)(Hom ‘𝑂)(1st𝑦)) = ((1st𝑦)(Hom ‘𝐶)(1st𝑥))
120119xpeq1i 5715 . . . . . . 7 (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) = (((1st𝑦)(Hom ‘𝐶)(1st𝑥)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))
121118, 120eqtrdi 2791 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) = (((1st𝑦)(Hom ‘𝐶)(1st𝑥)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))))
122114, 121feq12d 6725 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → ((𝑥(2nd𝑀)𝑦):(𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦)⟶(((Homf𝐶)‘𝑥)(Hom ‘𝐷)((Homf𝐶)‘𝑦)) ↔ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))):(((1st𝑦)(Hom ‘𝐶)(1st𝑥)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))⟶(((Homf𝐶)‘𝑥)(Hom ‘𝐷)((Homf𝐶)‘𝑦))))
123110, 122mpbird 257 . . . 4 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → (𝑥(2nd𝑀)𝑦):(𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦)⟶(((Homf𝐶)‘𝑥)(Hom ‘𝐷)((Homf𝐶)‘𝑦)))
124 eqid 2735 . . . . . . . . . 10 (Id‘𝐶) = (Id‘𝐶)
1252ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → 𝐶 ∈ Cat)
12655adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (1st𝑥) ∈ (Base‘𝐶))
127126adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (1st𝑥) ∈ (Base‘𝐶))
12867adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (2nd𝑥) ∈ (Base‘𝐶))
129128adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (2nd𝑥) ∈ (Base‘𝐶))
130 simpr 484 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
1313, 4, 124, 125, 127, 5, 129, 130catlid 17728 . . . . . . . . 9 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓) = 𝑓)
132131oveq1d 7446 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → ((((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓)(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥))) = (𝑓(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥))))
1333, 4, 124, 125, 127, 5, 129, 130catrid 17729 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (𝑓(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥))) = 𝑓)
134132, 133eqtrd 2775 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → ((((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓)(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥))) = 𝑓)
135134mpteq2dva 5248 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ ((((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓)(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥)))) = (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ 𝑓))
136 df-ov 7434 . . . . . . 7 (((Id‘𝐶)‘(1st𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)((Id‘𝐶)‘(2nd𝑥))) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)‘⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
1372adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝐶 ∈ Cat)
1383, 4, 124, 137, 126catidcl 17727 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(1st𝑥)) ∈ ((1st𝑥)(Hom ‘𝐶)(1st𝑥)))
1393, 4, 124, 137, 128catidcl 17727 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(2nd𝑥)) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑥)))
1401, 137, 3, 4, 126, 128, 126, 128, 5, 138, 139hof2val 18313 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (((Id‘𝐶)‘(1st𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)((Id‘𝐶)‘(2nd𝑥))) = (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ ((((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓)(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥)))))
141136, 140eqtr3id 2789 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)‘⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩) = (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ ((((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓)(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥)))))
14262adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
143142fveq2d 6911 . . . . . . . . . 10 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
144143, 90eqtr4di 2793 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Homf𝐶)(2nd𝑥)))
14533, 3, 4, 126, 128homfval 17737 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
146144, 145eqtrd 2775 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
147146reseq2d 6000 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf𝐶)‘𝑥)) = ( I ↾ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))))
148 mptresid 6071 . . . . . . 7 ( I ↾ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) = (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ 𝑓)
149147, 148eqtrdi 2791 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf𝐶)‘𝑥)) = (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ 𝑓))
150135, 141, 1493eqtr4d 2785 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)‘⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩) = ( I ↾ ((Homf𝐶)‘𝑥)))
151142, 142oveq12d 7449 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(2nd𝑀)𝑥) = (⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩))
152142fveq2d 6911 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ((Id‘(𝑂 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩))
15327adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑂 ∈ Cat)
154 eqid 2735 . . . . . . . 8 (Id‘𝑂) = (Id‘𝑂)
15515, 153, 137, 17, 3, 154, 124, 22, 126, 128xpcid 18245 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩) = ⟨((Id‘𝑂)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
15616, 124oppcid 17768 . . . . . . . . . 10 (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
157137, 156syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (Id‘𝑂) = (Id‘𝐶))
158157fveq1d 6909 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝑂)‘(1st𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
159158opeq1d 4884 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ⟨((Id‘𝑂)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩ = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
160152, 155, 1593eqtrd 2779 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
161151, 160fveq12d 6914 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((𝑥(2nd𝑀)𝑥)‘((Id‘(𝑂 ×c 𝐶))‘𝑥)) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)‘⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩))
16229adantr 480 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑈𝑉)
16338ffvelcdmda 7104 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) ∈ 𝑈)
16430, 23, 162, 163setcid 18140 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐷)‘((Homf𝐶)‘𝑥)) = ( I ↾ ((Homf𝐶)‘𝑥)))
165150, 161, 1643eqtr4d 2785 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((𝑥(2nd𝑀)𝑥)‘((Id‘(𝑂 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((Homf𝐶)‘𝑥)))
16623ad2ant1 1132 . . . . . 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)
167293ad2ant1 1132 . . . . . 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))) → 𝑈𝑉)
168363ad2ant1 1132 . . . . . 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 (Homf𝐶) ⊆ 𝑈)
169 simp21 1205 . . . . . . 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)))
170169, 55syl 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))) → (1st𝑥) ∈ (Base‘𝐶))
171169, 67syl 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))) → (2nd𝑥) ∈ (Base‘𝐶))
172 simp22 1206 . . . . . . 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)))
173172, 51syl 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))) → (1st𝑦) ∈ (Base‘𝐶))
174172, 58syl 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))) → (2nd𝑦) ∈ (Base‘𝐶))
175 simp23 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))) → 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)))
176 xp1st 8045 . . . . . . 7 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st𝑧) ∈ (Base‘𝐶))
177175, 176syl 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))) → (1st𝑧) ∈ (Base‘𝐶))
178 xp2nd 8046 . . . . . . 7 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd𝑧) ∈ (Base‘𝐶))
179175, 178syl 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))) → (2nd𝑧) ∈ (Base‘𝐶))
180 simp3l 1200 . . . . . . . . 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))
18115, 18, 115, 4, 20, 169, 172xpchom 18236 . . . . . . . . 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) = (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))))
182180, 181eleqtrd 2841 . . . . . . . 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 ∈ (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))))
183 xp1st 8045 . . . . . . . 8 (𝑓 ∈ (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) → (1st𝑓) ∈ ((1st𝑥)(Hom ‘𝑂)(1st𝑦)))
184182, 183syl 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))) → (1st𝑓) ∈ ((1st𝑥)(Hom ‘𝑂)(1st𝑦)))
185184, 119eleqtrdi 2849 . . . . . 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))) → (1st𝑓) ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)))
186 xp2nd 8046 . . . . . . 7 (𝑓 ∈ (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) → (2nd𝑓) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))
187182, 186syl 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))) → (2nd𝑓) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))
188 simp3r 1201 . . . . . . . . 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))
18915, 18, 115, 4, 20, 172, 175xpchom 18236 . . . . . . . . 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) = (((1st𝑦)(Hom ‘𝑂)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))))
190188, 189eleqtrd 2841 . . . . . . . 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 ∈ (((1st𝑦)(Hom ‘𝑂)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))))
191 xp1st 8045 . . . . . . . 8 (𝑔 ∈ (((1st𝑦)(Hom ‘𝑂)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))) → (1st𝑔) ∈ ((1st𝑦)(Hom ‘𝑂)(1st𝑧)))
192190, 191syl 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))) → (1st𝑔) ∈ ((1st𝑦)(Hom ‘𝑂)(1st𝑧)))
1934, 16oppchom 17761 . . . . . . 7 ((1st𝑦)(Hom ‘𝑂)(1st𝑧)) = ((1st𝑧)(Hom ‘𝐶)(1st𝑦))
194192, 193eleqtrdi 2849 . . . . . 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))) → (1st𝑔) ∈ ((1st𝑧)(Hom ‘𝐶)(1st𝑦)))
195 xp2nd 8046 . . . . . . 7 (𝑔 ∈ (((1st𝑦)(Hom ‘𝑂)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))) → (2nd𝑔) ∈ ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧)))
196190, 195syl 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))) → (2nd𝑔) ∈ ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧)))
1971, 16, 30, 166, 167, 168, 3, 4, 170, 171, 173, 174, 177, 179, 185, 187, 194, 196hofcllem 18315 . . . . 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))) → (((1st𝑓)(⟨(1st𝑧), (1st𝑦)⟩(comp‘𝐶)(1st𝑥))(1st𝑔))(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐶)(2nd𝑧))(2nd𝑓))) = (((1st𝑔)(⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)(2nd𝑔))(⟨((1st𝑥)(Hom ‘𝐶)(2nd𝑥)), ((1st𝑦)(Hom ‘𝐶)(2nd𝑦))⟩(comp‘𝐷)((1st𝑧)(Hom ‘𝐶)(2nd𝑧)))((1st𝑓)(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)(2nd𝑓))))
198169, 62syl 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 = ⟨(1st𝑥), (2nd𝑥)⟩)
199 1st2nd2 8052 . . . . . . . . 9 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
200175, 199syl 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 = ⟨(1st𝑧), (2nd𝑧)⟩)
201198, 200oveq12d 7449 . . . . . . 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(2nd𝑀)𝑧) = (⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩))
202172, 79syl 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 = ⟨(1st𝑦), (2nd𝑦)⟩)
203198, 202opeq12d 4886 . . . . . . . . . 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⟩ = ⟨⟨(1st𝑥), (2nd𝑥)⟩, ⟨(1st𝑦), (2nd𝑦)⟩⟩)
204203, 200oveq12d 7449 . . . . . . . . 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) = (⟨⟨(1st𝑥), (2nd𝑥)⟩, ⟨(1st𝑦), (2nd𝑦)⟩⟩(comp‘(𝑂 ×c 𝐶))⟨(1st𝑧), (2nd𝑧)⟩))
205 1st2nd2 8052 . . . . . . . . . 10 (𝑔 ∈ (((1st𝑦)(Hom ‘𝑂)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
206190, 205syl 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 = ⟨(1st𝑔), (2nd𝑔)⟩)
207 1st2nd2 8052 . . . . . . . . . 10 (𝑓 ∈ (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
208182, 207syl 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 = ⟨(1st𝑓), (2nd𝑓)⟩)
209204, 206, 208oveq123d 7452 . . . . . . . 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) = (⟨(1st𝑔), (2nd𝑔)⟩(⟨⟨(1st𝑥), (2nd𝑥)⟩, ⟨(1st𝑦), (2nd𝑦)⟩⟩(comp‘(𝑂 ×c 𝐶))⟨(1st𝑧), (2nd𝑧)⟩)⟨(1st𝑓), (2nd𝑓)⟩))
210 eqid 2735 . . . . . . . . 9 (comp‘𝑂) = (comp‘𝑂)
21115, 17, 3, 115, 4, 170, 171, 173, 174, 210, 5, 24, 177, 179, 184, 187, 192, 196xpcco2 18243 . . . . . . . 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))) → (⟨(1st𝑔), (2nd𝑔)⟩(⟨⟨(1st𝑥), (2nd𝑥)⟩, ⟨(1st𝑦), (2nd𝑦)⟩⟩(comp‘(𝑂 ×c 𝐶))⟨(1st𝑧), (2nd𝑧)⟩)⟨(1st𝑓), (2nd𝑓)⟩) = ⟨((1st𝑔)(⟨(1st𝑥), (1st𝑦)⟩(comp‘𝑂)(1st𝑧))(1st𝑓)), ((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐶)(2nd𝑧))(2nd𝑓))⟩)
2123, 5, 16, 170, 173, 177oppcco 17763 . . . . . . . . 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))) → ((1st𝑔)(⟨(1st𝑥), (1st𝑦)⟩(comp‘𝑂)(1st𝑧))(1st𝑓)) = ((1st𝑓)(⟨(1st𝑧), (1st𝑦)⟩(comp‘𝐶)(1st𝑥))(1st𝑔)))
213212opeq1d 4884 . . . . . . . 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))) → ⟨((1st𝑔)(⟨(1st𝑥), (1st𝑦)⟩(comp‘𝑂)(1st𝑧))(1st𝑓)), ((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐶)(2nd𝑧))(2nd𝑓))⟩ = ⟨((1st𝑓)(⟨(1st𝑧), (1st𝑦)⟩(comp‘𝐶)(1st𝑥))(1st𝑔)), ((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐶)(2nd𝑧))(2nd𝑓))⟩)
214209, 211, 2133eqtrd 2779 . . . . . . 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) = ⟨((1st𝑓)(⟨(1st𝑧), (1st𝑦)⟩(comp‘𝐶)(1st𝑥))(1st𝑔)), ((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐶)(2nd𝑧))(2nd𝑓))⟩)
215201, 214fveq12d 6914 . . . . . 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(2nd𝑀)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑂 ×c 𝐶))𝑧)𝑓)) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨((1st𝑓)(⟨(1st𝑧), (1st𝑦)⟩(comp‘𝐶)(1st𝑥))(1st𝑔)), ((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐶)(2nd𝑧))(2nd𝑓))⟩))
216 df-ov 7434 . . . . . 6 (((1st𝑓)(⟨(1st𝑧), (1st𝑦)⟩(comp‘𝐶)(1st𝑥))(1st𝑔))(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐶)(2nd𝑧))(2nd𝑓))) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨((1st𝑓)(⟨(1st𝑧), (1st𝑦)⟩(comp‘𝐶)(1st𝑥))(1st𝑔)), ((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐶)(2nd𝑧))(2nd𝑓))⟩)
217215, 216eqtr4di 2793 . . . . 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(2nd𝑀)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑂 ×c 𝐶))𝑧)𝑓)) = (((1st𝑓)(⟨(1st𝑧), (1st𝑦)⟩(comp‘𝐶)(1st𝑥))(1st𝑔))(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)((2nd𝑔)(⟨(2nd𝑥), (2nd𝑦)⟩(comp‘𝐶)(2nd𝑧))(2nd𝑓))))
218198fveq2d 6911 . . . . . . . . . 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))) → ((Homf𝐶)‘𝑥) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
219218, 90eqtr4di 2793 . . . . . . . . 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))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Homf𝐶)(2nd𝑥)))
22033, 3, 4, 170, 171homfval 17737 . . . . . . . . 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))) → ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
221219, 220eqtrd 2775 . . . . . . . 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))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
222202fveq2d 6911 . . . . . . . . . 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))) → ((Homf𝐶)‘𝑦) = ((Homf𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩))
223222, 98eqtr4di 2793 . . . . . . . . 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))) → ((Homf𝐶)‘𝑦) = ((1st𝑦)(Homf𝐶)(2nd𝑦)))
22433, 3, 4, 173, 174homfval 17737 . . . . . . . . 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))) → ((1st𝑦)(Homf𝐶)(2nd𝑦)) = ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)))
225223, 224eqtrd 2775 . . . . . . . 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))) → ((Homf𝐶)‘𝑦) = ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)))
226221, 225opeq12d 4886 . . . . . . 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))) → ⟨((Homf𝐶)‘𝑥), ((Homf𝐶)‘𝑦)⟩ = ⟨((1st𝑥)(Hom ‘𝐶)(2nd𝑥)), ((1st𝑦)(Hom ‘𝐶)(2nd𝑦))⟩)
227200fveq2d 6911 . . . . . . . . 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))) → ((Homf𝐶)‘𝑧) = ((Homf𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩))
228 df-ov 7434 . . . . . . . . 9 ((1st𝑧)(Homf𝐶)(2nd𝑧)) = ((Homf𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩)
229227, 228eqtr4di 2793 . . . . . . . 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))) → ((Homf𝐶)‘𝑧) = ((1st𝑧)(Homf𝐶)(2nd𝑧)))
23033, 3, 4, 177, 179homfval 17737 . . . . . . . 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))) → ((1st𝑧)(Homf𝐶)(2nd𝑧)) = ((1st𝑧)(Hom ‘𝐶)(2nd𝑧)))
231229, 230eqtrd 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))) → ((Homf𝐶)‘𝑧) = ((1st𝑧)(Hom ‘𝐶)(2nd𝑧)))
232226, 231oveq12d 7449 . . . . . 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))) → (⟨((Homf𝐶)‘𝑥), ((Homf𝐶)‘𝑦)⟩(comp‘𝐷)((Homf𝐶)‘𝑧)) = (⟨((1st𝑥)(Hom ‘𝐶)(2nd𝑥)), ((1st𝑦)(Hom ‘𝐶)(2nd𝑦))⟩(comp‘𝐷)((1st𝑧)(Hom ‘𝐶)(2nd𝑧))))
233202, 200oveq12d 7449 . . . . . . . 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(2nd𝑀)𝑧) = (⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩))
234233, 206fveq12d 6914 . . . . . . 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(2nd𝑀)𝑧)‘𝑔) = ((⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩))
235 df-ov 7434 . . . . . . 7 ((1st𝑔)(⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)(2nd𝑔)) = ((⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
236234, 235eqtr4di 2793 . . . . . 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(2nd𝑀)𝑧)‘𝑔) = ((1st𝑔)(⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)(2nd𝑔)))
237198, 202oveq12d 7449 . . . . . . . 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(2nd𝑀)𝑦) = (⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩))
238237, 208fveq12d 6914 . . . . . . 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(2nd𝑀)𝑦)‘𝑓) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)‘⟨(1st𝑓), (2nd𝑓)⟩))
239 df-ov 7434 . . . . . . 7 ((1st𝑓)(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)(2nd𝑓)) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)‘⟨(1st𝑓), (2nd𝑓)⟩)
240238, 239eqtr4di 2793 . . . . . 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(2nd𝑀)𝑦)‘𝑓) = ((1st𝑓)(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)(2nd𝑓)))
241232, 236, 240oveq123d 7452 . . . . 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(2nd𝑀)𝑧)‘𝑔)(⟨((Homf𝐶)‘𝑥), ((Homf𝐶)‘𝑦)⟩(comp‘𝐷)((Homf𝐶)‘𝑧))((𝑥(2nd𝑀)𝑦)‘𝑓)) = (((1st𝑔)(⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)(2nd𝑔))(⟨((1st𝑥)(Hom ‘𝐶)(2nd𝑥)), ((1st𝑦)(Hom ‘𝐶)(2nd𝑦))⟩(comp‘𝐷)((1st𝑧)(Hom ‘𝐶)(2nd𝑧)))((1st𝑓)(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)(2nd𝑓))))
242197, 217, 2413eqtr4d 2785 . . . 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(2nd𝑀)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑂 ×c 𝐶))𝑧)𝑓)) = (((𝑦(2nd𝑀)𝑧)‘𝑔)(⟨((Homf𝐶)‘𝑥), ((Homf𝐶)‘𝑦)⟩(comp‘𝐷)((Homf𝐶)‘𝑧))((𝑥(2nd𝑀)𝑦)‘𝑓)))
24318, 19, 20, 21, 22, 23, 24, 25, 28, 32, 41, 48, 123, 165, 242isfuncd 17916 . . 3 (𝜑 → (Homf𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd𝑀))
244 df-br 5149 . . 3 ((Homf𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd𝑀) ↔ ⟨(Homf𝐶), (2nd𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
245243, 244sylib 218 . 2 (𝜑 → ⟨(Homf𝐶), (2nd𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
24614, 245eqeltrd 2839 1 (𝜑𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  cop 4637   class class class wbr 5148  cmpt 5231   I cid 5582   × cxp 5687  ran crn 5690  cres 5691   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710  Homf chomf 17711  oppCatcoppc 17756   Func cfunc 17905  SetCatcsetc 18129   ×c cxpc 18224  HomFchof 18305
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-homf 17715  df-oppc 17757  df-func 17909  df-setc 18130  df-xpc 18228  df-hof 18307
This theorem is referenced by:  oppchofcl  18317  oppcyon  18326  yonedalem1  18329  yonedalem21  18330  yonedalem22  18335
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