MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hofcl Structured version   Visualization version   GIF version

Theorem hofcl 17342
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 2797 . . . 4 (Base‘𝐶) = (Base‘𝐶)
4 eqid 2797 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2797 . . . 4 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5hofval 17335 . . 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 6558 . . . . . 6 (Homf𝐶) ∈ V
8 fvex 6558 . . . . . . . 8 (Base‘𝐶) ∈ V
98, 8xpex 7340 . . . . . . 7 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
109, 9mpoex 7640 . . . . . 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 7563 . . . . 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 4723 . . 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 2836 . 2 (𝜑𝑀 = ⟨(Homf𝐶), (2nd𝑀)⟩)
15 eqid 2797 . . . . 5 (𝑂 ×c 𝐶) = (𝑂 ×c 𝐶)
16 hofcl.o . . . . . 6 𝑂 = (oppCat‘𝐶)
1716, 3oppcbas 16821 . . . . 5 (Base‘𝐶) = (Base‘𝑂)
1815, 17, 3xpcbas 17261 . . . 4 ((Base‘𝐶) × (Base‘𝐶)) = (Base‘(𝑂 ×c 𝐶))
19 eqid 2797 . . . 4 (Base‘𝐷) = (Base‘𝐷)
20 eqid 2797 . . . 4 (Hom ‘(𝑂 ×c 𝐶)) = (Hom ‘(𝑂 ×c 𝐶))
21 eqid 2797 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
22 eqid 2797 . . . 4 (Id‘(𝑂 ×c 𝐶)) = (Id‘(𝑂 ×c 𝐶))
23 eqid 2797 . . . 4 (Id‘𝐷) = (Id‘𝐷)
24 eqid 2797 . . . 4 (comp‘(𝑂 ×c 𝐶)) = (comp‘(𝑂 ×c 𝐶))
25 eqid 2797 . . . 4 (comp‘𝐷) = (comp‘𝐷)
2616oppccat 16825 . . . . . 6 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
272, 26syl 17 . . . . 5 (𝜑𝑂 ∈ Cat)
2815, 27, 2xpccat 17273 . . . 4 (𝜑 → (𝑂 ×c 𝐶) ∈ Cat)
29 hofcl.u . . . . 5 (𝜑𝑈𝑉)
30 hofcl.d . . . . . 6 𝐷 = (SetCat‘𝑈)
3130setccat 17178 . . . . 5 (𝑈𝑉𝐷 ∈ Cat)
3229, 31syl 17 . . . 4 (𝜑𝐷 ∈ Cat)
33 eqid 2797 . . . . . . . 8 (Homf𝐶) = (Homf𝐶)
3433, 3homffn 16796 . . . . . . 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 6236 . . . . . 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 17171 . . . . . 6 (𝜑𝑈 = (Base‘𝐷))
4039feq3d 6376 . . . . 5 (𝜑 → ((Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶𝑈 ↔ (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷)))
4138, 40mpbid 233 . . . 4 (𝜑 → (Homf𝐶):((Base‘𝐶) × (Base‘𝐶))⟶(Base‘𝐷))
42 eqid 2797 . . . . . 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 7055 . . . . . . 7 ((1st𝑦)(Hom ‘𝐶)(1st𝑥)) ∈ V
44 ovex 7055 . . . . . . 7 ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ∈ V
4543, 44mpoex 7640 . . . . . 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 7631 . . . . 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 6323 . . . . 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 259 . . . 4 (𝜑 → (2nd𝑀) Fn (((Base‘𝐶) × (Base‘𝐶)) × ((Base‘𝐶) × (Base‘𝐶))))
492ad3antrrr 726 . . . . . . . . . . . 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 774 . . . . . . . . . . . . . 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 7584 . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . 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 7584 . . . . . . . . . . . . . 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 7585 . . . . . . . . . . . . . 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 773 . . . . . . . . . . . 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 7591 . . . . . . . . . . . . . . . . 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 7038 . . . . . . . . . . . . . 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 7040 . . . . . . . . . . . . 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 7585 . . . . . . . . . . . . . . . 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 6549 . . . . . . . . . . . . . . . . 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 7026 . . . . . . . . . . . . . . . . 17 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
7270, 71syl6eqr 2851 . . . . . . . . . . . . . . . 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 2870 . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . 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 16789 . . . . . . . . . . . . 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 2885 . . . . . . . . . . . 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 16789 . . . . . . . . . . 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 7591 . . . . . . . . . . . . . . 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 6549 . . . . . . . . . . . . 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 7026 . . . . . . . . . . . . 13 ((1st𝑦)(Hom ‘𝐶)(2nd𝑦)) = ((Hom ‘𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩)
8381, 82syl6eqr 2851 . . . . . . . . . . . 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 2888 . . . . . . . . . 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 6749 . . . . . . . . 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 722 . . . . . . . . . 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 16795 . . . . . . . . . . . 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 6549 . . . . . . . . . . . . 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 7026 . . . . . . . . . . . . 13 ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
9189, 90syl6eqr 2851 . . . . . . . . . . . 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 2843 . . . . . . . . . . 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 722 . . . . . . . . . . . 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, 54ffvelrnd 6724 . . . . . . . . . . 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 2886 . . . . . . . . . 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 16795 . . . . . . . . . . . 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 6549 . . . . . . . . . . . . 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 7026 . . . . . . . . . . . . 13 ((1st𝑦)(Homf𝐶)(2nd𝑦)) = ((Homf𝐶)‘⟨(1st𝑦), (2nd𝑦)⟩)
9997, 98syl6eqr 2851 . . . . . . . . . . . 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 2843 . . . . . . . . . . 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, 50ffvelrnd 6724 . . . . . . . . . . 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 2886 . . . . . . . . . 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 17174 . . . . . . . . 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 258 . . . . . . . 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 7041 . . . . . . . 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 2888 . . . . . . 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 3160 . . . . . 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 2797 . . . . . . 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 7629 . . . . . 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 219 . . . . 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 7040 . . . . . . 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 7160 . . . . . . . 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 1442 . . . . . . 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 2853 . . . . . 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 2797 . . . . . . . 8 (Hom ‘𝑂) = (Hom ‘𝑂)
116 simprl 767 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)))
117 simprr 769 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)))
11815, 18, 115, 4, 20, 116, 117xpchom 17263 . . . . . . 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 16818 . . . . . . . 8 ((1st𝑥)(Hom ‘𝑂)(1st𝑦)) = ((1st𝑦)(Hom ‘𝐶)(1st𝑥))
120119xpeq1i 5476 . . . . . . 7 (((1st𝑥)(Hom ‘𝑂)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) = (((1st𝑦)(Hom ‘𝐶)(1st𝑥)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)))
121118, 120syl6eq 2849 . . . . . 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 6377 . . . . 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 258 . . . 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 2797 . . . . . . . . . 10 (Id‘𝐶) = (Id‘𝐶)
1252ad2antrr 722 . . . . . . . . . 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 16787 . . . . . . . . 9 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (((Id‘𝐶)‘(2nd𝑥))(⟨(1st𝑥), (2nd𝑥)⟩(comp‘𝐶)(2nd𝑥))𝑓) = 𝑓)
132131oveq1d 7038 . . . . . . . 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 16788 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) ∧ 𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))) → (𝑓(⟨(1st𝑥), (1st𝑥)⟩(comp‘𝐶)(2nd𝑥))((Id‘𝐶)‘(1st𝑥))) = 𝑓)
134132, 133eqtrd 2833 . . . . . . 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 5062 . . . . . 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 7026 . . . . . . 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 16786 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(1st𝑥)) ∈ ((1st𝑥)(Hom ‘𝐶)(1st𝑥)))
1393, 4, 124, 137, 128catidcl 16786 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐶)‘(2nd𝑥)) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑥)))
1401, 137, 3, 4, 126, 128, 126, 128, 5, 138, 139hof2val 17339 . . . . . . 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, 140syl5eqr 2847 . . . . . 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 6549 . . . . . . . . . 10 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((Homf𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
144143, 90syl6eqr 2851 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Homf𝐶)(2nd𝑥)))
14533, 3, 4, 126, 128homfval 16795 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((1st𝑥)(Homf𝐶)(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
146144, 145eqtrd 2833 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
147146reseq2d 5741 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf𝐶)‘𝑥)) = ( I ↾ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥))))
148 mptresid 5805 . . . . . . 7 (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ 𝑓) = ( I ↾ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
149147, 148syl6eqr 2851 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ( I ↾ ((Homf𝐶)‘𝑥)) = (𝑓 ∈ ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) ↦ 𝑓))
150135, 141, 1493eqtr4d 2843 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩)‘⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩) = ( I ↾ ((Homf𝐶)‘𝑥)))
151142, 142oveq12d 7041 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (𝑥(2nd𝑀)𝑥) = (⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑥), (2nd𝑥)⟩))
152142fveq2d 6549 . . . . . . 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 2797 . . . . . . . 8 (Id‘𝑂) = (Id‘𝑂)
15515, 153, 137, 17, 3, 154, 124, 22, 126, 128xpcid 17272 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘⟨(1st𝑥), (2nd𝑥)⟩) = ⟨((Id‘𝑂)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
15616, 124oppcid 16824 . . . . . . . . . 10 (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
157137, 156syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → (Id‘𝑂) = (Id‘𝐶))
158157fveq1d 6547 . . . . . . . 8 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝑂)‘(1st𝑥)) = ((Id‘𝐶)‘(1st𝑥)))
159158opeq1d 4722 . . . . . . 7 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ⟨((Id‘𝑂)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩ = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
160152, 155, 1593eqtrd 2837 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘(𝑂 ×c 𝐶))‘𝑥) = ⟨((Id‘𝐶)‘(1st𝑥)), ((Id‘𝐶)‘(2nd𝑥))⟩)
161151, 160fveq12d 6552 . . . . 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)
16338ffvelrnda 6723 . . . . . 6 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Homf𝐶)‘𝑥) ∈ 𝑈)
16430, 23, 162, 163setcid 17179 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((Id‘𝐷)‘((Homf𝐶)‘𝑥)) = ( I ↾ ((Homf𝐶)‘𝑥)))
165150, 161, 1643eqtr4d 2843 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶))) → ((𝑥(2nd𝑀)𝑥)‘((Id‘(𝑂 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((Homf𝐶)‘𝑥)))
16623ad2ant1 1126 . . . . . 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 1126 . . . . . 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 1126 . . . . . 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 1199 . . . . . . 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 1200 . . . . . . 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 1201 . . . . . . 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 7584 . . . . . . 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 7585 . . . . . . 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 1194 . . . . . . . . 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 17263 . . . . . . . . 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 2887 . . . . . . . 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 7584 . . . . . . . 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, 119syl6eleq 2895 . . . . . 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 7585 . . . . . . 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 1195 . . . . . . . . 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 17263 . . . . . . . . 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 2887 . . . . . . . 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 7584 . . . . . . . 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 16818 . . . . . . 7 ((1st𝑦)(Hom ‘𝑂)(1st𝑧)) = ((1st𝑧)(Hom ‘𝐶)(1st𝑦))
194192, 193syl6eleq 2895 . . . . . 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 7585 . . . . . . 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 17341 . . . . 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 7591 . . . . . . . . 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 7041 . . . . . . 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 4724 . . . . . . . . . 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 7041 . . . . . . . . 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 7591 . . . . . . . . . 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 7591 . . . . . . . . . 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 7044 . . . . . . . 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 2797 . . . . . . . . 9 (comp‘𝑂) = (comp‘𝑂)
21115, 17, 3, 115, 4, 170, 171, 173, 174, 210, 5, 24, 177, 179, 184, 187, 192, 196xpcco2 17270 . . . . . . . 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 16820 . . . . . . . . 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 4722 . . . . . . . 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 2837 . . . . . . 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 6552 . . . . . 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 7026 . . . . . 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, 216syl6eqr 2851 . . . . 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 6549 . . . . . . . . . 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, 90syl6eqr 2851 . . . . . . . . 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 16795 . . . . . . . . 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 2833 . . . . . . . 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 6549 . . . . . . . . . 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, 98syl6eqr 2851 . . . . . . . . 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 16795 . . . . . . . . 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 2833 . . . . . . . 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 4724 . . . . . . 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 6549 . . . . . . . . 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 7026 . . . . . . . . 9 ((1st𝑧)(Homf𝐶)(2nd𝑧)) = ((Homf𝐶)‘⟨(1st𝑧), (2nd𝑧)⟩)
229227, 228syl6eqr 2851 . . . . . . . 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 16795 . . . . . . . 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 2833 . . . . . . 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 7041 . . . . . 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 7041 . . . . . . . 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 6552 . . . . . . 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 7026 . . . . . . 7 ((1st𝑔)(⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)(2nd𝑔)) = ((⟨(1st𝑦), (2nd𝑦)⟩(2nd𝑀)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
236234, 235syl6eqr 2851 . . . . . 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 7041 . . . . . . . 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 6552 . . . . . . 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 7026 . . . . . . 7 ((1st𝑓)(⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)(2nd𝑓)) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝑀)⟨(1st𝑦), (2nd𝑦)⟩)‘⟨(1st𝑓), (2nd𝑓)⟩)
240238, 239syl6eqr 2851 . . . . . 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 7044 . . . . 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 2843 . . . 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 16968 . . 3 (𝜑 → (Homf𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd𝑀))
244 df-br 4969 . . 3 ((Homf𝐶)((𝑂 ×c 𝐶) Func 𝐷)(2nd𝑀) ↔ ⟨(Homf𝐶), (2nd𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
245243, 244sylib 219 . 2 (𝜑 → ⟨(Homf𝐶), (2nd𝑀)⟩ ∈ ((𝑂 ×c 𝐶) Func 𝐷))
24614, 245eqeltrd 2885 1 (𝜑𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1525  wcel 2083  wral 3107  Vcvv 3440  wss 3865  cop 4484   class class class wbr 4968  cmpt 5047   I cid 5354   × cxp 5448  ran crn 5451  cres 5452   Fn wfn 6227  wf 6228  cfv 6232  (class class class)co 7023  cmpo 7025  1st c1st 7550  2nd c2nd 7551  Basecbs 16316  Hom chom 16409  compcco 16410  Catccat 16768  Idccid 16769  Homf chomf 16770  oppCatcoppc 16814   Func cfunc 16957  SetCatcsetc 17168   ×c cxpc 17251  HomFchof 17331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-tpos 7750  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-map 8265  df-ixp 8318  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-3 11555  df-4 11556  df-5 11557  df-6 11558  df-7 11559  df-8 11560  df-9 11561  df-n0 11752  df-z 11836  df-dec 11953  df-uz 12098  df-fz 12747  df-struct 16318  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-hom 16422  df-cco 16423  df-cat 16772  df-cid 16773  df-homf 16774  df-oppc 16815  df-func 16961  df-setc 17169  df-xpc 17255  df-hof 17333
This theorem is referenced by:  oppchofcl  17343  oppcyon  17352  yonedalem1  17355  yonedalem21  17356  yonedalem22  17361
  Copyright terms: Public domain W3C validator