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Theorem hofcl 18148
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 18141 . . 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 6855 . . . . . 6 (Homf𝐶) ∈ V
8 fvex 6855 . . . . . . . 8 (Base‘𝐶) ∈ V
98, 8xpex 7687 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
109, 9mpoex 8012 . . . . . 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 7932 . . . . 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 4837 . . 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 17599 . . . . 5 (Base‘𝐶) = (Base‘𝑂)
1815, 17, 3xpcbas 18066 . . . 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 17604 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
272, 26syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
2815, 27, 2xpccat 18078 . . . 4 (𝜑 → (𝑂 ×c 𝐶) ∈ Cat)
29 hofcl.u . . . . 5 (𝜑𝑈𝑉)
30 hofcl.d . . . . . 6 𝐷 = (SetCat‘𝑈)
3130setccat 17971 . . . . 5 (𝑈𝑉𝐷 ∈ Cat)
3229, 31syl 17 . . . 4 (𝜑𝐷 ∈ Cat)
33 eqid 2736 . . . . . . . 8 (Homf𝐶) = (Homf𝐶)
3433, 3homffn 17573 . . . . . . 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 6500 . . . . . 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 17964 . . . . . 6 (𝜑𝑈 = (Base‘𝐷))
4039feq3d 6655 . . . . 5 (𝜑 → ((Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷)))
4138, 40mpbid 231 . . . 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 7390 . . . . . . 7 ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∈ V
44 ovex 7390 . . . . . . 7 ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ∈ V
4543, 44mpoex 8012 . . . . . 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 8002 . . . . 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 6595 . . . . 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 257 . . . 4 (𝜑 → (2nd𝑀) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶))))
492ad3antrrr 728 . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 7953 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 775 . . . . . . . . . . . . . 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 7953 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 7954 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 775 . . . . . . . . . . . 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 7960 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 7372 . . . . . . . . . . . . . 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 7374 . . . . . . . . . . . . 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 7954 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . . . 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 7360 . . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 17565 . . . . . . . . . . . . 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 2838 . . . . . . . . . . . 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 17565 . . . . . . . . . . 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 7960 . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . 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 7360 . . . . . . . . . . . . 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 481 . . . . . . . . . . 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 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)) → ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓) ∈ ((Hom ‘𝐶)‘𝑦))
8685fmpttd 7063 . . . . . . . . 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 724 . . . . . . . . . 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 17572 . . . . . . . . . . . 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 6846 . . . . . . . . . . . . 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 7360 . . . . . . . . . . . . 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 724 . . . . . . . . . . . 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 7036 . . . . . . . . . . 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 2839 . . . . . . . . . 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 17572 . . . . . . . . . . . 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 6846 . . . . . . . . . . . . 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 7360 . . . . . . . . . . . . 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 7036 . . . . . . . . . . 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 2839 . . . . . . . . . 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 17967 . . . . . . . . 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 256 . . . . . . . 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 7375 . . . . . . . 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 2841 . . . . . . 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 3197 . . . . . 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 8000 . . . . . 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 217 . . . . 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 7374 . . . . . . 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 7502 . . . . . . . 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 1450 . . . . . . 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 769 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
117 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
11815, 18, 115, 4, 20, 116, 117xpchom 18068 . . . . . . 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 17596 . . . . . . . 8 ((1st𝑥)(Hom ‘𝑂)(1st𝑦)) = ((1st𝑦)(Hom ‘𝐶)(1st𝑥))
120119xpeq1i 5659 . . . . . . 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 6656 . . . . 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 256 . . . 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 724 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → 𝐶 ∈ Cat)
12655adantl 482 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (1st𝑥) ∈ (Base‘𝐶))
127126adantr 481 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (1st𝑥) ∈ (Base‘𝐶))
12867adantl 482 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (2nd𝑥) ∈ (Base‘𝐶))
129128adantr 481 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (2nd𝑥) ∈ (Base‘𝐶))
130 simpr 485 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
1313, 4, 124, 125, 127, 5, 129, 130catlid 17563 . . . . . . . . 9 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓) = 𝑓)
132131oveq1d 7372 . . . . . . . 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 17564 . . . . . . . 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 5205 . . . . . 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 7360 . . . . . . 7 (((Id‘𝐶)‘(1st𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)((Id‘𝐶)‘(2nd𝑥))) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)‘⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
1372adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝐶 ∈ Cat)
1383, 4, 124, 137, 126catidcl 17562 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(1st𝑥)) ∈ ((1st𝑥)(Hom ‘𝐶)(1st𝑥)))
1393, 4, 124, 137, 128catidcl 17562 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(2nd𝑥)) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑥)))
1401, 137, 3, 4, 126, 128, 126, 128, 5, 138, 139hof2val 18145 . . . . . . 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 482 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
143142fveq2d 6846 . . . . . . . . . 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 17572 . . . . . . . . 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 5937 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf𝐶)‘𝑥)) = ( I ↾ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))))
148 mptresid 6004 . . . . . . 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 7375 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(2nd𝑀)𝑥) = (⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩))
152142fveq2d 6846 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ((Id‘(𝑂 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩))
15327adantr 481 . . . . . . . 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 18077 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩) = ⟨((Id‘𝑂)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
15616, 124oppcid 17603 . . . . . . . . . 10 (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
157137, 156syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (Id‘𝑂) = (Id‘𝐶))
158157fveq1d 6844 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝑂)‘(1st𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
159158opeq1d 4836 . . . . . . 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 6849 . . . . 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 481 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑈𝑉)
16338ffvelcdmda 7035 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) ∈ 𝑈)
16430, 23, 162, 163setcid 17972 . . . . 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 7953 . . . . . . 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 7954 . . . . . . 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 18068 . . . . . . . . 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 2840 . . . . . . . 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 7953 . . . . . . . 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 2848 . . . . . 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 7954 . . . . . . 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 18068 . . . . . . . . 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 2840 . . . . . . . 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 7953 . . . . . . . 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 17596 . . . . . . 7 ((1st𝑦)(Hom ‘𝑂)(1st𝑧)) = ((1st𝑧)(Hom ‘𝐶)(1st𝑦))
194192, 193eleqtrdi 2848 . . . . . 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 7954 . . . . . . 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 18147 . . . . 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 7960 . . . . . . . . 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 7375 . . . . . . 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 4838 . . . . . . . . . 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 7375 . . . . . . . . 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 7960 . . . . . . . . . 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 7960 . . . . . . . . . 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 7378 . . . . . . . 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 18075 . . . . . . . 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 17598 . . . . . . . . 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 4836 . . . . . . . 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 6849 . . . . . 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 7360 . . . . . 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 6846 . . . . . . . . . 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 17572 . . . . . . . . 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 6846 . . . . . . . . . 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 17572 . . . . . . . . 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 4838 . . . . . . 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 6846 . . . . . . . . 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 7360 . . . . . . . . 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 17572 . . . . . . . 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 7375 . . . . . 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 7375 . . . . . . . 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 6849 . . . . . . 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 7360 . . . . . . 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 7375 . . . . . . . 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 6849 . . . . . . 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 7360 . . . . . . 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 7378 . . . . 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 17751 . . 3 (𝜑 → (Homf𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd𝑀))
244 df-br 5106 . . 3 ((Homf𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd𝑀) ↔ ⟨(Homf𝐶), (2nd𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
245243, 244sylib 217 . 2 (𝜑 → ⟨(Homf𝐶), (2nd𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
24614, 245eqeltrd 2838 1 (𝜑𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  wss 3910  cop 4592   class class class wbr 5105  cmpt 5188   I cid 5530   × cxp 5631  ran crn 5634  cres 5635   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Idccid 17545  Homf chomf 17546  oppCatcoppc 17591   Func cfunc 17740  SetCatcsetc 17961   ×c cxpc 18056  HomFchof 18137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-homf 17550  df-oppc 17592  df-func 17744  df-setc 17962  df-xpc 18060  df-hof 18139
This theorem is referenced by:  oppchofcl  18149  oppcyon  18158  yonedalem1  18161  yonedalem21  18162  yonedalem22  18167
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