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Theorem hofcl 18305
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 2736 . . . 4 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2736 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2736 . . . 4 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5hofval 18298 . . 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 6918 . . . . . 6 (Homf𝐶) ∈ V
8 fvex 6918 . . . . . . . 8 (Base‘𝐶) ∈ V
98, 8xpex 7774 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
109, 9mpoex 8105 . . . . . 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 8026 . . . . 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 4879 . . 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 2779 . 2 (𝜑𝑀 = ⟨(Homf𝐶), (2nd𝑀)⟩)
15 eqid 2736 . . . . 5 (𝑂 ×c 𝐶) = (𝑂 ×c 𝐶)
16 hofcl.o . . . . . 6 𝑂 = (oppCat‘𝐶)
1716, 3oppcbas 17762 . . . . 5 (Base‘𝐶) = (Base‘𝑂)
1815, 17, 3xpcbas 18224 . . . 4 ((Base‘𝐶) × (Base‘𝐶)) = (Base‘(𝑂 ×c 𝐶))
19 eqid 2736 . . . 4 (Base‘𝐷) = (Base‘𝐷)
20 eqid 2736 . . . 4 (Hom ‘(𝑂 ×c 𝐶)) = (Hom ‘(𝑂 ×c 𝐶))
21 eqid 2736 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
22 eqid 2736 . . . 4 (Id‘(𝑂 ×c 𝐶)) = (Id‘(𝑂 ×c 𝐶))
23 eqid 2736 . . . 4 (Id‘𝐷) = (Id‘𝐷)
24 eqid 2736 . . . 4 (comp‘(𝑂 ×c 𝐶)) = (comp‘(𝑂 ×c 𝐶))
25 eqid 2736 . . . 4 (comp‘𝐷) = (comp‘𝐷)
2616oppccat 17766 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
272, 26syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
2815, 27, 2xpccat 18236 . . . 4 (𝜑 → (𝑂 ×c 𝐶) ∈ Cat)
29 hofcl.u . . . . 5 (𝜑𝑈𝑉)
30 hofcl.d . . . . . 6 𝐷 = (SetCat‘𝑈)
3130setccat 18131 . . . . 5 (𝑈𝑉𝐷 ∈ Cat)
3229, 31syl 17 . . . 4 (𝜑𝐷 ∈ Cat)
33 eqid 2736 . . . . . . . 8 (Homf𝐶) = (Homf𝐶)
3433, 3homffn 17737 . . . . . . 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 6564 . . . . . 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 18124 . . . . . 6 (𝜑𝑈 = (Base‘𝐷))
4039feq3d 6722 . . . . 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 2736 . . . . . 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 7465 . . . . . . 7 ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∈ V
44 ovex 7465 . . . . . . 7 ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ∈ V
4543, 44mpoex 8105 . . . . . 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 8096 . . . . 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 6660 . . . . 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 777 . . . . . . . . . . . . . 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 8047 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 8047 . . . . . . . . . . . . . 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 8048 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . 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 8054 . . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . 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 8048 . . . . . . . . . . . . . . . 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 6909 . . . . . . . . . . . . . . . . 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 7435 . . . . . . . . . . . . . . . . 17 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
7270, 71eqtr4di 2794 . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . 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 17729 . . . . . . . . . . . . 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 2840 . . . . . . . . . . . 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 17729 . . . . . . . . . . 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 8054 . . . . . . . . . . . . . . 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 6909 . . . . . . . . . . . . 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 7435 . . . . . . . . . . . . 13 ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)) = ((Hom ‘𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩)
8381, 82eqtr4di 2794 . . . . . . . . . . . 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 2843 . . . . . . . . . 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 7134 . . . . . . . . 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 17736 . . . . . . . . . . . 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 6909 . . . . . . . . . . . . 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 7435 . . . . . . . . . . . . 13 ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
9189, 90eqtr4di 2794 . . . . . . . . . . . 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 2786 . . . . . . . . . . 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 7104 . . . . . . . . . . 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 2841 . . . . . . . . . 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 17736 . . . . . . . . . . . 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 6909 . . . . . . . . . . . . 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 7435 . . . . . . . . . . . . 13 ((1st𝑦)(Homf𝐶)(2nd𝑦)) = ((Homf𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩)
9997, 98eqtr4di 2794 . . . . . . . . . . . 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 2786 . . . . . . . . . . 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 7104 . . . . . . . . . . 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 2841 . . . . . . . . . 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 18127 . . . . . . . . 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 7450 . . . . . . . 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 2843 . . . . . . 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 3201 . . . . . 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 2736 . . . . . . 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 8094 . . . . . 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 7449 . . . . . . 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 7581 . . . . . . . 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 1451 . . . . . . 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 2796 . . . . . 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 2736 . . . . . . . 8 (Hom ‘𝑂) = (Hom ‘𝑂)
116 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
117 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
11815, 18, 115, 4, 20, 116, 117xpchom 18226 . . . . . . 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 17759 . . . . . . . 8 ((1st𝑥)(Hom ‘𝑂)(1st𝑦)) = ((1st𝑦)(Hom ‘𝐶)(1st𝑥))
120119xpeq1i 5710 . . . . . . 7 (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) = (((1st𝑦)(Hom ‘𝐶)(1st𝑥)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))
121118, 120eqtrdi 2792 . . . . . 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 6723 . . . . 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 2736 . . . . . . . . . 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 17727 . . . . . . . . 9 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓) = 𝑓)
132131oveq1d 7447 . . . . . . . 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 17728 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (𝑓(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥))) = 𝑓)
134132, 133eqtrd 2776 . . . . . . 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 5241 . . . . . 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 7435 . . . . . . 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 17726 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(1st𝑥)) ∈ ((1st𝑥)(Hom ‘𝐶)(1st𝑥)))
1393, 4, 124, 137, 128catidcl 17726 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(2nd𝑥)) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑥)))
1401, 137, 3, 4, 126, 128, 126, 128, 5, 138, 139hof2val 18302 . . . . . . 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 2790 . . . . . 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 6909 . . . . . . . . . 10 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
144143, 90eqtr4di 2794 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Homf𝐶)(2nd𝑥)))
14533, 3, 4, 126, 128homfval 17736 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
146144, 145eqtrd 2776 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
147146reseq2d 5996 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf𝐶)‘𝑥)) = ( I ↾ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))))
148 mptresid 6068 . . . . . . 7 ( I ↾ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) = (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ 𝑓)
149147, 148eqtrdi 2792 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf𝐶)‘𝑥)) = (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ 𝑓))
150135, 141, 1493eqtr4d 2786 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)‘⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩) = ( I ↾ ((Homf𝐶)‘𝑥)))
151142, 142oveq12d 7450 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(2nd𝑀)𝑥) = (⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩))
152142fveq2d 6909 . . . . . . 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 2736 . . . . . . . 8 (Id‘𝑂) = (Id‘𝑂)
15515, 153, 137, 17, 3, 154, 124, 22, 126, 128xpcid 18235 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩) = ⟨((Id‘𝑂)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
15616, 124oppcid 17765 . . . . . . . . . 10 (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
157137, 156syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (Id‘𝑂) = (Id‘𝐶))
158157fveq1d 6907 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝑂)‘(1st𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
159158opeq1d 4878 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ⟨((Id‘𝑂)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩ = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
160152, 155, 1593eqtrd 2780 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
161151, 160fveq12d 6912 . . . . 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 7103 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) ∈ 𝑈)
16430, 23, 162, 163setcid 18132 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐷)‘((Homf𝐶)‘𝑥)) = ( I ↾ ((Homf𝐶)‘𝑥)))
165150, 161, 1643eqtr4d 2786 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((𝑥(2nd𝑀)𝑥)‘((Id‘(𝑂 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((Homf𝐶)‘𝑥)))
16623ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝐶 ∈ Cat)
167293ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑈𝑉)
168363ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → ran (Homf𝐶) ⊆ 𝑈)
169 simp21 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))) → 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 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))) → 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 1208 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)))
176 xp1st 8047 . . . . . . 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 8048 . . . . . . 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 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))) → 𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦))
18115, 18, 115, 4, 20, 169, 172xpchom 18226 . . . . . . . . 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 2842 . . . . . . . 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 8047 . . . . . . . 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 2850 . . . . . 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 8048 . . . . . . 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 1202 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑂 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(𝑂 ×c 𝐶))𝑧))
18915, 18, 115, 4, 20, 172, 175xpchom 18226 . . . . . . . . 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 2842 . . . . . . . 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 8047 . . . . . . . 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 17759 . . . . . . 7 ((1st𝑦)(Hom ‘𝑂)(1st𝑧)) = ((1st𝑧)(Hom ‘𝐶)(1st𝑦))
194192, 193eleqtrdi 2850 . . . . . 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 8048 . . . . . . 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 18304 . . . . 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 8054 . . . . . . . . 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 7450 . . . . . . 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 4880 . . . . . . . . . 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 7450 . . . . . . . . 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 8054 . . . . . . . . . 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 8054 . . . . . . . . . 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 7453 . . . . . . . 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 2736 . . . . . . . . 9 (comp‘𝑂) = (comp‘𝑂)
21115, 17, 3, 115, 4, 170, 171, 173, 174, 210, 5, 24, 177, 179, 184, 187, 192, 196xpcco2 18233 . . . . . . . 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 17761 . . . . . . . . 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 4878 . . . . . . . 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 2780 . . . . . . 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 6912 . . . . . 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 7435 . . . . . 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 2794 . . . . 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 6909 . . . . . . . . . 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 2794 . . . . . . . . 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 17736 . . . . . . . . 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 2776 . . . . . . . 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 6909 . . . . . . . . . 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 2794 . . . . . . . . 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 17736 . . . . . . . . 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 2776 . . . . . . . 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 4880 . . . . . . 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 6909 . . . . . . . . 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 7435 . . . . . . . . 9 ((1st𝑧)(Homf𝐶)(2nd𝑧)) = ((Homf𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩)
229227, 228eqtr4di 2794 . . . . . . . 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 17736 . . . . . . . 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 2776 . . . . . . 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 7450 . . . . . 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 7450 . . . . . . . 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 6912 . . . . . . 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 7435 . . . . . . 7 ((1st𝑔)(⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)(2nd𝑔)) = ((⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
236234, 235eqtr4di 2794 . . . . . 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 7450 . . . . . . . 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 6912 . . . . . . 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 7435 . . . . . . 7 ((1st𝑓)(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)(2nd𝑓)) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)‘⟨(1st𝑓), (2nd𝑓)⟩)
240238, 239eqtr4di 2794 . . . . . 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 7453 . . . . 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 2786 . . . 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 17911 . . 3 (𝜑 → (Homf𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd𝑀))
244 df-br 5143 . . 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 2840 1 (𝜑𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  wss 3950  cop 4631   class class class wbr 5142  cmpt 5224   I cid 5576   × cxp 5682  ran crn 5685  cres 5686   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  1st c1st 8013  2nd c2nd 8014  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709  Homf chomf 17710  oppCatcoppc 17755   Func cfunc 17900  SetCatcsetc 18121   ×c cxpc 18214  HomFchof 18294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-cat 17712  df-cid 17713  df-homf 17714  df-oppc 17756  df-func 17904  df-setc 18122  df-xpc 18218  df-hof 18296
This theorem is referenced by:  oppchofcl  18306  oppcyon  18315  yonedalem1  18318  yonedalem21  18319  yonedalem22  18324
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