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Theorem hofcl 16722
 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 2610 . . . 4 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2610 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2610 . . . 4 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5hofval 16715 . . 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 6113 . . . . . 6 (Homf𝐶) ∈ V
8 fvex 6113 . . . . . . . 8 (Base‘𝐶) ∈ V
98, 8xpex 6860 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
109, 9mpt2ex 7136 . . . . . 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 7070 . . . . 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 4347 . . 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 2647 . 2 (𝜑𝑀 = ⟨(Homf𝐶), (2nd𝑀)⟩)
15 eqid 2610 . . . . 5 (𝑂 ×c 𝐶) = (𝑂 ×c 𝐶)
16 hofcl.o . . . . . 6 𝑂 = (oppCat‘𝐶)
1716, 3oppcbas 16201 . . . . 5 (Base‘𝐶) = (Base‘𝑂)
1815, 17, 3xpcbas 16641 . . . 4 ((Base‘𝐶) × (Base‘𝐶)) = (Base‘(𝑂 ×c 𝐶))
19 eqid 2610 . . . 4 (Base‘𝐷) = (Base‘𝐷)
20 eqid 2610 . . . 4 (Hom ‘(𝑂 ×c 𝐶)) = (Hom ‘(𝑂 ×c 𝐶))
21 eqid 2610 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
22 eqid 2610 . . . 4 (Id‘(𝑂 ×c 𝐶)) = (Id‘(𝑂 ×c 𝐶))
23 eqid 2610 . . . 4 (Id‘𝐷) = (Id‘𝐷)
24 eqid 2610 . . . 4 (comp‘(𝑂 ×c 𝐶)) = (comp‘(𝑂 ×c 𝐶))
25 eqid 2610 . . . 4 (comp‘𝐷) = (comp‘𝐷)
2616oppccat 16205 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
272, 26syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
2815, 27, 2xpccat 16653 . . . 4 (𝜑 → (𝑂 ×c 𝐶) ∈ Cat)
29 hofcl.u . . . . 5 (𝜑𝑈𝑉)
30 hofcl.d . . . . . 6 𝐷 = (SetCat‘𝑈)
3130setccat 16558 . . . . 5 (𝑈𝑉𝐷 ∈ Cat)
3229, 31syl 17 . . . 4 (𝜑𝐷 ∈ Cat)
33 eqid 2610 . . . . . . . 8 (Homf𝐶) = (Homf𝐶)
3433, 3homffn 16176 . . . . . . 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 5808 . . . . . 6 ((Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ ((Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ran (Homf𝐶) ⊆ 𝑈))
3835, 36, 37sylanbrc 695 . . . . 5 (𝜑 → (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈)
3930, 29setcbas 16551 . . . . . 6 (𝜑𝑈 = (Base‘𝐷))
4039feq3d 5945 . . . . 5 (𝜑 → ((Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷)))
4138, 40mpbid 221 . . . 4 (𝜑 → (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷))
42 eqid 2610 . . . . . 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 6577 . . . . . . 7 ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∈ V
44 ovex 6577 . . . . . . 7 ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ∈ V
4543, 44mpt2ex 7136 . . . . . 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, 45fnmpt2i 7128 . . . . 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 5895 . . . . 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 247 . . . 4 (𝜑 → (2nd𝑀) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶))))
492ad3antrrr 762 . . . . . . . . . . . 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 797 . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . 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 796 . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . 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 7090 . . . . . . . . . . . . . 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 796 . . . . . . . . . . . 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 7096 . . . . . . . . . . . . . . . . 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 6564 . . . . . . . . . . . . . 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 6566 . . . . . . . . . . . . 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 7090 . . . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . . . . . 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 6552 . . . . . . . . . . . . . . . . 17 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
7270, 71syl6eqr 2662 . . . . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . 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 797 . . . . . . . . . . . . . 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 16169 . . . . . . . . . . . . 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 2688 . . . . . . . . . . . 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 16169 . . . . . . . . . . 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 7096 . . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . 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 6552 . . . . . . . . . . . . 13 ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)) = ((Hom ‘𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩)
8381, 82syl6eqr 2662 . . . . . . . . . . . 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 2691 . . . . . . . . . 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))
86 eqid 2610 . . . . . . . . . 10 ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓)) = ( ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝐶)(2nd𝑦))𝑓))
8785, 86fmptd 6292 . . . . . . . . 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))
8829ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) ∧ (𝑓 ∈ ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∧ 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))) → 𝑈𝑉)
8933, 3, 4, 56, 68homfval 16175 . . . . . . . . . . . 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)))
9063fveq2d 6107 . . . . . . . . . . . . 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)⟩))
91 df-ov 6552 . . . . . . . . . . . . 13 ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
9290, 91syl6eqr 2662 . . . . . . . . . . . 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)))
9389, 92, 723eqtr4d 2654 . . . . . . . . . . 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))
9438ad2antrr 758 . . . . . . . . . . . 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)
9594, 54ffvelrnd 6268 . . . . . . . . . . 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)
9693, 95eqeltrrd 2689 . . . . . . . . . 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)
9733, 3, 4, 52, 59homfval 16175 . . . . . . . . . . . 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)))
9880fveq2d 6107 . . . . . . . . . . . . 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)⟩))
99 df-ov 6552 . . . . . . . . . . . . 13 ((1st𝑦)(Homf𝐶)(2nd𝑦)) = ((Homf𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩)
10098, 99syl6eqr 2662 . . . . . . . . . . . 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)))
10197, 100, 833eqtr4d 2654 . . . . . . . . . . 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))
10294, 50ffvelrnd 6268 . . . . . . . . . . 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)
103101, 102eqeltrrd 2689 . . . . . . . . . 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)
10430, 88, 21, 96, 103elsetchom 16554 . . . . . . . . 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)))
10587, 104mpbird 246 . . . . . . . 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)))
10693, 101oveq12d 6567 . . . . . . . 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)))
107105, 106eleqtrrd 2691 . . . . . . 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)))
108107ralrimivva 2954 . . . . . 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)))
109 eqid 2610 . . . . . . 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)))
110109fmpt2 7126 . . . . . 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)))
111108, 110sylib 207 . . . . 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)))
11212oveqd 6566 . . . . . . 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))
11342ovmpt4g 6681 . . . . . . . 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))))
11445, 113mp3an3 1405 . . . . . . 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))))
115112, 114sylan9eq 2664 . . . . . 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))))
116 eqid 2610 . . . . . . . 8 (Hom ‘𝑂) = (Hom ‘𝑂)
117 simprl 790 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
118 simprr 792 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
11915, 18, 116, 4, 20, 117, 118xpchom 16643 . . . . . . 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))))
1204, 16oppchom 16198 . . . . . . . 8 ((1st𝑥)(Hom ‘𝑂)(1st𝑦)) = ((1st𝑦)(Hom ‘𝐶)(1st𝑥))
121120xpeq1i 5059 . . . . . . 7 (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) = (((1st𝑦)(Hom ‘𝐶)(1st𝑥)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))
122119, 121syl6eq 2660 . . . . . 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))))
123115, 122feq12d 5946 . . . . 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))))
124111, 123mpbird 246 . . . 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)))
125 eqid 2610 . . . . . . . . . 10 (Id‘𝐶) = (Id‘𝐶)
1262ad2antrr 758 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → 𝐶 ∈ Cat)
12755adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (1st𝑥) ∈ (Base‘𝐶))
128127adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (1st𝑥) ∈ (Base‘𝐶))
12967adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (2nd𝑥) ∈ (Base‘𝐶))
130129adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (2nd𝑥) ∈ (Base‘𝐶))
131 simpr 476 . . . . . . . . . 10 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
1323, 4, 125, 126, 128, 5, 130, 131catlid 16167 . . . . . . . . 9 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓) = 𝑓)
133132oveq1d 6564 . . . . . . . 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))))
1343, 4, 125, 126, 128, 5, 130, 131catrid 16168 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (𝑓(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥))) = 𝑓)
135133, 134eqtrd 2644 . . . . . . 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)
136135mpteq2dva 4672 . . . . . 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))
137 df-ov 6552 . . . . . . 7 (((Id‘𝐶)‘(1st𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)((Id‘𝐶)‘(2nd𝑥))) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)‘⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
1382adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝐶 ∈ Cat)
1393, 4, 125, 138, 127catidcl 16166 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(1st𝑥)) ∈ ((1st𝑥)(Hom ‘𝐶)(1st𝑥)))
1403, 4, 125, 138, 129catidcl 16166 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(2nd𝑥)) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑥)))
1411, 138, 3, 4, 127, 129, 127, 129, 5, 139, 140hof2val 16719 . . . . . . 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)))))
142137, 141syl5eqr 2658 . . . . . 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)))))
14362adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
144143fveq2d 6107 . . . . . . . . . 10 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
145144, 91syl6eqr 2662 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Homf𝐶)(2nd𝑥)))
14633, 3, 4, 127, 129homfval 16175 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
147145, 146eqtrd 2644 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
148147reseq2d 5317 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf𝐶)‘𝑥)) = ( I ↾ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))))
149 mptresid 5375 . . . . . . 7 (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ 𝑓) = ( I ↾ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
150148, 149syl6eqr 2662 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf𝐶)‘𝑥)) = (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ 𝑓))
151136, 142, 1503eqtr4d 2654 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)‘⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩) = ( I ↾ ((Homf𝐶)‘𝑥)))
152143, 143oveq12d 6567 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(2nd𝑀)𝑥) = (⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩))
153143fveq2d 6107 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ((Id‘(𝑂 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩))
15427adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑂 ∈ Cat)
155 eqid 2610 . . . . . . . 8 (Id‘𝑂) = (Id‘𝑂)
15615, 154, 138, 17, 3, 155, 125, 22, 127, 129xpcid 16652 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩) = ⟨((Id‘𝑂)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
15716, 125oppcid 16204 . . . . . . . . . 10 (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
158138, 157syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (Id‘𝑂) = (Id‘𝐶))
159158fveq1d 6105 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝑂)‘(1st𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
160159opeq1d 4346 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ⟨((Id‘𝑂)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩ = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
161153, 156, 1603eqtrd 2648 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
162152, 161fveq12d 6109 . . . . 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))⟩))
16329adantr 480 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → 𝑈𝑉)
16438ffvelrnda 6267 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) ∈ 𝑈)
16530, 23, 163, 164setcid 16559 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐷)‘((Homf𝐶)‘𝑥)) = ( I ↾ ((Homf𝐶)‘𝑥)))
166151, 162, 1653eqtr4d 2654 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((𝑥(2nd𝑀)𝑥)‘((Id‘(𝑂 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((Homf𝐶)‘𝑥)))
16723ad2ant1 1075 . . . . . 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)
168293ad2ant1 1075 . . . . . 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)
169363ad2ant1 1075 . . . . . 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)
170 simp21 1087 . . . . . . 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)))
171170, 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))
172170, 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))
173 simp22 1088 . . . . . . 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)))
174173, 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))
175173, 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))
176 simp23 1089 . . . . . . 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)))
177 xp1st 7089 . . . . . . 7 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (1st𝑧) ∈ (Base‘𝐶))
178176, 177syl 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))
179 xp2nd 7090 . . . . . . 7 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → (2nd𝑧) ∈ (Base‘𝐶))
180176, 179syl 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))
181 simp3l 1082 . . . . . . . . 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))
18215, 18, 116, 4, 20, 170, 173xpchom 16643 . . . . . . . . 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))))
183181, 182eleqtrd 2690 . . . . . . . 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))))
184 xp1st 7089 . . . . . . . 8 (𝑓 ∈ (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) → (1st𝑓) ∈ ((1st𝑥)(Hom ‘𝑂)(1st𝑦)))
185183, 184syl 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)))
186185, 120syl6eleq 2698 . . . . . 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)))
187 xp2nd 7090 . . . . . . 7 (𝑓 ∈ (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) → (2nd𝑓) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))
188183, 187syl 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)))
189 simp3r 1083 . . . . . . . . 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))
19015, 18, 116, 4, 20, 173, 176xpchom 16643 . . . . . . . . 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))))
191189, 190eleqtrd 2690 . . . . . . . 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))))
192 xp1st 7089 . . . . . . . 8 (𝑔 ∈ (((1st𝑦)(Hom ‘𝑂)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))) → (1st𝑔) ∈ ((1st𝑦)(Hom ‘𝑂)(1st𝑧)))
193191, 192syl 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)))
1944, 16oppchom 16198 . . . . . . 7 ((1st𝑦)(Hom ‘𝑂)(1st𝑧)) = ((1st𝑧)(Hom ‘𝐶)(1st𝑦))
195193, 194syl6eleq 2698 . . . . . 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)))
196 xp2nd 7090 . . . . . . 7 (𝑔 ∈ (((1st𝑦)(Hom ‘𝑂)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))) → (2nd𝑔) ∈ ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧)))
197191, 196syl 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)))
1981, 16, 30, 167, 168, 169, 3, 4, 171, 172, 174, 175, 178, 180, 186, 188, 195, 197hofcllem 16721 . . . . 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))))
199170, 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)⟩)
200 1st2nd2 7096 . . . . . . . . 9 (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
201176, 200syl 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)⟩)
202199, 201oveq12d 6567 . . . . . . 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)⟩))
203173, 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)⟩)
204199, 203opeq12d 4348 . . . . . . . . . 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)⟩⟩)
205204, 201oveq12d 6567 . . . . . . . . 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)⟩))
206 1st2nd2 7096 . . . . . . . . . 10 (𝑔 ∈ (((1st𝑦)(Hom ‘𝑂)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
207191, 206syl 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)⟩)
208 1st2nd2 7096 . . . . . . . . . 10 (𝑓 ∈ (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
209183, 208syl 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)⟩)
210205, 207, 209oveq123d 6570 . . . . . . . 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)⟩))
211 eqid 2610 . . . . . . . . 9 (comp‘𝑂) = (comp‘𝑂)
21215, 17, 3, 116, 4, 171, 172, 174, 175, 211, 5, 24, 178, 180, 185, 188, 193, 197xpcco2 16650 . . . . . . . 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))⟩)
2133, 5, 16, 171, 174, 178oppcco 16200 . . . . . . . . 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)))
214213opeq1d 4346 . . . . . . . 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))⟩)
215210, 212, 2143eqtrd 2648 . . . . . . 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))⟩)
216202, 215fveq12d 6109 . . . . . 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))⟩))
217 df-ov 6552 . . . . . 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))⟩)
218216, 217syl6eqr 2662 . . . . 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))))
219199fveq2d 6107 . . . . . . . . . 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)⟩))
220219, 91syl6eqr 2662 . . . . . . . . 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)))
22133, 3, 4, 171, 172homfval 16175 . . . . . . . . 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)))
222220, 221eqtrd 2644 . . . . . . . 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)))
223203fveq2d 6107 . . . . . . . . . 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)⟩))
224223, 99syl6eqr 2662 . . . . . . . . 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)))
22533, 3, 4, 174, 175homfval 16175 . . . . . . . . 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)))
226224, 225eqtrd 2644 . . . . . . . 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)))
227222, 226opeq12d 4348 . . . . . . 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))⟩)
228201fveq2d 6107 . . . . . . . . 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)⟩))
229 df-ov 6552 . . . . . . . . 9 ((1st𝑧)(Homf𝐶)(2nd𝑧)) = ((Homf𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩)
230228, 229syl6eqr 2662 . . . . . . . 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)))
23133, 3, 4, 178, 180homfval 16175 . . . . . . . 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)))
232230, 231eqtrd 2644 . . . . . . 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)))
233227, 232oveq12d 6567 . . . . . 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))))
234203, 201oveq12d 6567 . . . . . . . 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)⟩))
235234, 207fveq12d 6109 . . . . . . 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)⟩))
236 df-ov 6552 . . . . . . 7 ((1st𝑔)(⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)(2nd𝑔)) = ((⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
237235, 236syl6eqr 2662 . . . . . 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)))
238199, 203oveq12d 6567 . . . . . . . 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)⟩))
239238, 209fveq12d 6109 . . . . . . 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)⟩))
240 df-ov 6552 . . . . . . 7 ((1st𝑓)(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)(2nd𝑓)) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)‘⟨(1st𝑓), (2nd𝑓)⟩)
241239, 240syl6eqr 2662 . . . . . 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)))
242233, 237, 241oveq123d 6570 . . . . 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))))
243198, 218, 2423eqtr4d 2654 . . . 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)))
24418, 19, 20, 21, 22, 23, 24, 25, 28, 32, 41, 48, 124, 166, 243isfuncd 16348 . . 3 (𝜑 → (Homf𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd𝑀))
245 df-br 4584 . . 3 ((Homf𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd𝑀) ↔ ⟨(Homf𝐶), (2nd𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
246244, 245sylib 207 . 2 (𝜑 → ⟨(Homf𝐶), (2nd𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
24714, 246eqeltrd 2688 1 (𝜑𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   × cxp 5036  ran crn 5039   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  Homf chomf 16150  oppCatcoppc 16194   Func cfunc 16337  SetCatcsetc 16548   ×c cxpc 16631  HomFchof 16711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-homf 16154  df-oppc 16195  df-func 16341  df-setc 16549  df-xpc 16635  df-hof 16713 This theorem is referenced by:  oppchofcl  16723  oppcyon  16732  yonedalem1  16735  yonedalem21  16736  yonedalem22  16741
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