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Theorem evlfcl 18268
Description: The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors 𝐶𝐷, and the second parameter in 𝐷. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfcl.e 𝐸 = (𝐶 evalF 𝐷)
evlfcl.q 𝑄 = (𝐶 FuncCat 𝐷)
evlfcl.c (𝜑𝐶 ∈ Cat)
evlfcl.d (𝜑𝐷 ∈ Cat)
Assertion
Ref Expression
evlfcl (𝜑𝐸 ∈ ((𝑄 ×c 𝐶) Func 𝐷))

Proof of Theorem evlfcl
Dummy variables 𝑓 𝑎 𝑔 𝑚 𝑛 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlfcl.e . . . . 5 𝐸 = (𝐶 evalF 𝐷)
2 evlfcl.c . . . . 5 (𝜑𝐶 ∈ Cat)
3 evlfcl.d . . . . 5 (𝜑𝐷 ∈ Cat)
4 eqid 2736 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2736 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2736 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
7 eqid 2736 . . . . 5 (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
81, 2, 3, 4, 5, 6, 7evlfval 18263 . . . 4 (𝜑𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)
9 ovex 7465 . . . . . 6 (𝐶 Func 𝐷) ∈ V
10 fvex 6918 . . . . . 6 (Base‘𝐶) ∈ V
119, 10mpoex 8105 . . . . 5 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)) ∈ V
129, 10xpex 7774 . . . . . 6 ((𝐶 Func 𝐷) × (Base‘𝐶)) ∈ V
1312, 12mpoex 8105 . . . . 5 (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))) ∈ V
1411, 13opelvv 5724 . . . 4 ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩ ∈ (V × V)
158, 14eqeltrdi 2848 . . 3 (𝜑𝐸 ∈ (V × V))
16 1st2nd2 8054 . . 3 (𝐸 ∈ (V × V) → 𝐸 = ⟨(1st𝐸), (2nd𝐸)⟩)
1715, 16syl 17 . 2 (𝜑𝐸 = ⟨(1st𝐸), (2nd𝐸)⟩)
18 eqid 2736 . . . . 5 (𝑄 ×c 𝐶) = (𝑄 ×c 𝐶)
19 evlfcl.q . . . . . 6 𝑄 = (𝐶 FuncCat 𝐷)
2019fucbas 18009 . . . . 5 (𝐶 Func 𝐷) = (Base‘𝑄)
2118, 20, 4xpcbas 18224 . . . 4 ((𝐶 Func 𝐷) × (Base‘𝐶)) = (Base‘(𝑄 ×c 𝐶))
22 eqid 2736 . . . 4 (Base‘𝐷) = (Base‘𝐷)
23 eqid 2736 . . . 4 (Hom ‘(𝑄 ×c 𝐶)) = (Hom ‘(𝑄 ×c 𝐶))
24 eqid 2736 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
25 eqid 2736 . . . 4 (Id‘(𝑄 ×c 𝐶)) = (Id‘(𝑄 ×c 𝐶))
26 eqid 2736 . . . 4 (Id‘𝐷) = (Id‘𝐷)
27 eqid 2736 . . . 4 (comp‘(𝑄 ×c 𝐶)) = (comp‘(𝑄 ×c 𝐶))
2819, 2, 3fuccat 18019 . . . . 5 (𝜑𝑄 ∈ Cat)
2918, 28, 2xpccat 18236 . . . 4 (𝜑 → (𝑄 ×c 𝐶) ∈ Cat)
30 relfunc 17908 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
31 simpr 484 . . . . . . . . . . 11 ((𝜑𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
32 1st2ndbr 8068 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (1st𝑓)(𝐶 Func 𝐷)(2nd𝑓))
3330, 31, 32sylancr 587 . . . . . . . . . 10 ((𝜑𝑓 ∈ (𝐶 Func 𝐷)) → (1st𝑓)(𝐶 Func 𝐷)(2nd𝑓))
344, 22, 33funcf1 17912 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐶 Func 𝐷)) → (1st𝑓):(Base‘𝐶)⟶(Base‘𝐷))
3534ffvelcdmda 7103 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐶 Func 𝐷)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st𝑓)‘𝑥) ∈ (Base‘𝐷))
3635ralrimiva 3145 . . . . . . 7 ((𝜑𝑓 ∈ (𝐶 Func 𝐷)) → ∀𝑥 ∈ (Base‘𝐶)((1st𝑓)‘𝑥) ∈ (Base‘𝐷))
3736ralrimiva 3145 . . . . . 6 (𝜑 → ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑥 ∈ (Base‘𝐶)((1st𝑓)‘𝑥) ∈ (Base‘𝐷))
38 eqid 2736 . . . . . . 7 (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥))
3938fmpo 8094 . . . . . 6 (∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑥 ∈ (Base‘𝐶)((1st𝑓)‘𝑥) ∈ (Base‘𝐷) ↔ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷))
4037, 39sylib 218 . . . . 5 (𝜑 → (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷))
4111, 13op1std 8025 . . . . . . 7 (𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩ → (1st𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)))
428, 41syl 17 . . . . . 6 (𝜑 → (1st𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)))
4342feq1d 6719 . . . . 5 (𝜑 → ((1st𝐸):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷) ↔ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷)))
4440, 43mpbird 257 . . . 4 (𝜑 → (1st𝐸):((𝐶 Func 𝐷) × (Base‘𝐶))⟶(Base‘𝐷))
45 eqid 2736 . . . . . 6 (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))) = (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))
46 ovex 7465 . . . . . . . . 9 (𝑚(𝐶 Nat 𝐷)𝑛) ∈ V
47 ovex 7465 . . . . . . . . 9 ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ∈ V
4846, 47mpoex 8105 . . . . . . . 8 (𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))) ∈ V
4948csbex 5310 . . . . . . 7 (1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))) ∈ V
5049csbex 5310 . . . . . 6 (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))) ∈ V
5145, 50fnmpoi 8096 . . . . 5 (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))) Fn (((𝐶 Func 𝐷) × (Base‘𝐶)) × ((𝐶 Func 𝐷) × (Base‘𝐶)))
5211, 13op2ndd 8026 . . . . . . 7 (𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (Base‘𝐶) ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩ → (2nd𝐸) = (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))))
538, 52syl 17 . . . . . 6 (𝜑 → (2nd𝐸) = (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))))
5453fneq1d 6660 . . . . 5 (𝜑 → ((2nd𝐸) Fn (((𝐶 Func 𝐷) × (Base‘𝐶)) × ((𝐶 Func 𝐷) × (Base‘𝐶))) ↔ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)), 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝐷)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔)))) Fn (((𝐶 Func 𝐷) × (Base‘𝐶)) × ((𝐶 Func 𝐷) × (Base‘𝐶)))))
5551, 54mpbiri 258 . . . 4 (𝜑 → (2nd𝐸) Fn (((𝐶 Func 𝐷) × (Base‘𝐶)) × ((𝐶 Func 𝐷) × (Base‘𝐶))))
563ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
5756adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝐷 ∈ Cat)
58 simplrl 776 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝐶 Func 𝐷))
5930, 58, 32sylancr 587 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (1st𝑓)(𝐶 Func 𝐷)(2nd𝑓))
604, 22, 59funcf1 17912 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (1st𝑓):(Base‘𝐶)⟶(Base‘𝐷))
6160adantr 480 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → (1st𝑓):(Base‘𝐶)⟶(Base‘𝐷))
62 simplrr 777 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝑢 ∈ (Base‘𝐶))
6362adantr 480 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝑢 ∈ (Base‘𝐶))
6461, 63ffvelcdmd 7104 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((1st𝑓)‘𝑢) ∈ (Base‘𝐷))
65 simplrr 777 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝑣 ∈ (Base‘𝐶))
6661, 65ffvelcdmd 7104 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((1st𝑓)‘𝑣) ∈ (Base‘𝐷))
67 simprl 770 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝑔 ∈ (𝐶 Func 𝐷))
68 1st2ndbr 8068 . . . . . . . . . . . . . . . . . . 19 ((Rel (𝐶 Func 𝐷) ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (1st𝑔)(𝐶 Func 𝐷)(2nd𝑔))
6930, 67, 68sylancr 587 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (1st𝑔)(𝐶 Func 𝐷)(2nd𝑔))
704, 22, 69funcf1 17912 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (1st𝑔):(Base‘𝐶)⟶(Base‘𝐷))
7170adantr 480 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → (1st𝑔):(Base‘𝐶)⟶(Base‘𝐷))
7271, 65ffvelcdmd 7104 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((1st𝑔)‘𝑣) ∈ (Base‘𝐷))
73 simprr 772 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝑣 ∈ (Base‘𝐶))
744, 5, 24, 59, 62, 73funcf2 17914 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (𝑢(2nd𝑓)𝑣):(𝑢(Hom ‘𝐶)𝑣)⟶(((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑓)‘𝑣)))
7574adantr 480 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → (𝑢(2nd𝑓)𝑣):(𝑢(Hom ‘𝐶)𝑣)⟶(((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑓)‘𝑣)))
76 simprr 772 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ∈ (𝑢(Hom ‘𝐶)𝑣))
7775, 76ffvelcdmd 7104 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((𝑢(2nd𝑓)𝑣)‘) ∈ (((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑓)‘𝑣)))
78 simprl 770 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔))
797, 78nat1st2nd 18000 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → 𝑎 ∈ (⟨(1st𝑓), (2nd𝑓)⟩(𝐶 Nat 𝐷)⟨(1st𝑔), (2nd𝑔)⟩))
807, 79, 4, 24, 65natcl 18002 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → (𝑎𝑣) ∈ (((1st𝑓)‘𝑣)(Hom ‘𝐷)((1st𝑔)‘𝑣)))
8122, 24, 6, 57, 64, 66, 72, 77, 80catcocl 17729 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) ∧ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔) ∧ ∈ (𝑢(Hom ‘𝐶)𝑣))) → ((𝑎𝑣)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑣)⟩(comp‘𝐷)((1st𝑔)‘𝑣))((𝑢(2nd𝑓)𝑣)‘)) ∈ (((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣)))
8281ralrimivva 3201 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → ∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀ ∈ (𝑢(Hom ‘𝐶)𝑣)((𝑎𝑣)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑣)⟩(comp‘𝐷)((1st𝑔)‘𝑣))((𝑢(2nd𝑓)𝑣)‘)) ∈ (((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣)))
83 eqid 2736 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎𝑣)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑣)⟩(comp‘𝐷)((1st𝑔)‘𝑣))((𝑢(2nd𝑓)𝑣)‘))) = (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎𝑣)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑣)⟩(comp‘𝐷)((1st𝑔)‘𝑣))((𝑢(2nd𝑓)𝑣)‘)))
8483fmpo 8094 . . . . . . . . . . . . 13 (∀𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔)∀ ∈ (𝑢(Hom ‘𝐶)𝑣)((𝑎𝑣)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑣)⟩(comp‘𝐷)((1st𝑔)‘𝑣))((𝑢(2nd𝑓)𝑣)‘)) ∈ (((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣)) ↔ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎𝑣)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑣)⟩(comp‘𝐷)((1st𝑔)‘𝑣))((𝑢(2nd𝑓)𝑣)‘))):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣)))
8582, 84sylib 218 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎𝑣)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑣)⟩(comp‘𝐷)((1st𝑔)‘𝑣))((𝑢(2nd𝑓)𝑣)‘))):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣)))
862ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
87 eqid 2736 . . . . . . . . . . . . . 14 (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩)
881, 86, 56, 4, 5, 6, 7, 58, 67, 62, 73, 87evlf2 18264 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩) = (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎𝑣)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑣)⟩(comp‘𝐷)((1st𝑔)‘𝑣))((𝑢(2nd𝑓)𝑣)‘))))
8988feq1d 6719 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣)) ↔ (𝑎 ∈ (𝑓(𝐶 Nat 𝐷)𝑔), ∈ (𝑢(Hom ‘𝐶)𝑣) ↦ ((𝑎𝑣)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑣)⟩(comp‘𝐷)((1st𝑔)‘𝑣))((𝑢(2nd𝑓)𝑣)‘))):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣))))
9085, 89mpbird 257 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣)))
9119, 7fuchom 18010 . . . . . . . . . . . . 13 (𝐶 Nat 𝐷) = (Hom ‘𝑄)
9218, 20, 4, 91, 5, 58, 62, 67, 73, 23xpchom2 18232 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩) = ((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣)))
931, 86, 56, 4, 58, 62evlf1 18266 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (𝑓(1st𝐸)𝑢) = ((1st𝑓)‘𝑢))
941, 86, 56, 4, 67, 73evlf1 18266 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (𝑔(1st𝐸)𝑣) = ((1st𝑔)‘𝑣))
9593, 94oveq12d 7450 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → ((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)) = (((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣)))
9692, 95feq23d 6730 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)) ↔ (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):((𝑓(𝐶 Nat 𝐷)𝑔) × (𝑢(Hom ‘𝐶)𝑣))⟶(((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑔)‘𝑣))))
9790, 96mpbird 257 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) ∧ (𝑔 ∈ (𝐶 Func 𝐷) ∧ 𝑣 ∈ (Base‘𝐶))) → (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)))
9897ralrimivva 3201 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)))
9998ralrimivva 3201 . . . . . . . 8 (𝜑 → ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑢 ∈ (Base‘𝐶)∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)))
100 oveq2 7440 . . . . . . . . . . . 12 (𝑦 = ⟨𝑔, 𝑣⟩ → (𝑥(2nd𝐸)𝑦) = (𝑥(2nd𝐸)⟨𝑔, 𝑣⟩))
101 oveq2 7440 . . . . . . . . . . . 12 (𝑦 = ⟨𝑔, 𝑣⟩ → (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) = (𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩))
102 fveq2 6905 . . . . . . . . . . . . . 14 (𝑦 = ⟨𝑔, 𝑣⟩ → ((1st𝐸)‘𝑦) = ((1st𝐸)‘⟨𝑔, 𝑣⟩))
103 df-ov 7435 . . . . . . . . . . . . . 14 (𝑔(1st𝐸)𝑣) = ((1st𝐸)‘⟨𝑔, 𝑣⟩)
104102, 103eqtr4di 2794 . . . . . . . . . . . . 13 (𝑦 = ⟨𝑔, 𝑣⟩ → ((1st𝐸)‘𝑦) = (𝑔(1st𝐸)𝑣))
105104oveq2d 7448 . . . . . . . . . . . 12 (𝑦 = ⟨𝑔, 𝑣⟩ → (((1st𝐸)‘𝑥)(Hom ‘𝐷)((1st𝐸)‘𝑦)) = (((1st𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)))
106100, 101, 105feq123d 6724 . . . . . . . . . . 11 (𝑦 = ⟨𝑔, 𝑣⟩ → ((𝑥(2nd𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)((1st𝐸)‘𝑦)) ↔ (𝑥(2nd𝐸)⟨𝑔, 𝑣⟩):(𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣))))
107106ralxp 5851 . . . . . . . . . 10 (∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)((1st𝐸)‘𝑦)) ↔ ∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(𝑥(2nd𝐸)⟨𝑔, 𝑣⟩):(𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)))
108 oveq1 7439 . . . . . . . . . . . 12 (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(2nd𝐸)⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩))
109 oveq1 7439 . . . . . . . . . . . 12 (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩) = (⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩))
110 fveq2 6905 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑓, 𝑢⟩ → ((1st𝐸)‘𝑥) = ((1st𝐸)‘⟨𝑓, 𝑢⟩))
111 df-ov 7435 . . . . . . . . . . . . . 14 (𝑓(1st𝐸)𝑢) = ((1st𝐸)‘⟨𝑓, 𝑢⟩)
112110, 111eqtr4di 2794 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑓, 𝑢⟩ → ((1st𝐸)‘𝑥) = (𝑓(1st𝐸)𝑢))
113112oveq1d 7447 . . . . . . . . . . . 12 (𝑥 = ⟨𝑓, 𝑢⟩ → (((1st𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)) = ((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)))
114108, 109, 113feq123d 6724 . . . . . . . . . . 11 (𝑥 = ⟨𝑓, 𝑢⟩ → ((𝑥(2nd𝐸)⟨𝑔, 𝑣⟩):(𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)) ↔ (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣))))
1151142ralbidv 3220 . . . . . . . . . 10 (𝑥 = ⟨𝑓, 𝑢⟩ → (∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(𝑥(2nd𝐸)⟨𝑔, 𝑣⟩):(𝑥(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)) ↔ ∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣))))
116107, 115bitrid 283 . . . . . . . . 9 (𝑥 = ⟨𝑓, 𝑢⟩ → (∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)((1st𝐸)‘𝑦)) ↔ ∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣))))
117116ralxp 5851 . . . . . . . 8 (∀𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)((1st𝐸)‘𝑦)) ↔ ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑢 ∈ (Base‘𝐶)∀𝑔 ∈ (𝐶 Func 𝐷)∀𝑣 ∈ (Base‘𝐶)(⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑔, 𝑣⟩):(⟨𝑓, 𝑢⟩(Hom ‘(𝑄 ×c 𝐶))⟨𝑔, 𝑣⟩)⟶((𝑓(1st𝐸)𝑢)(Hom ‘𝐷)(𝑔(1st𝐸)𝑣)))
11899, 117sylibr 234 . . . . . . 7 (𝜑 → ∀𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)((1st𝐸)‘𝑦)))
119118r19.21bi 3250 . . . . . 6 ((𝜑𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) → ∀𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))(𝑥(2nd𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)((1st𝐸)‘𝑦)))
120119r19.21bi 3250 . . . . 5 (((𝜑𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) → (𝑥(2nd𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)((1st𝐸)‘𝑦)))
121120anasss 466 . . . 4 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))) → (𝑥(2nd𝐸)𝑦):(𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦)⟶(((1st𝐸)‘𝑥)(Hom ‘𝐷)((1st𝐸)‘𝑦)))
12228adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝑄 ∈ Cat)
1232adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
124 eqid 2736 . . . . . . . . . . 11 (Id‘𝑄) = (Id‘𝑄)
125 eqid 2736 . . . . . . . . . . 11 (Id‘𝐶) = (Id‘𝐶)
126 simprl 770 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝐶 Func 𝐷))
127 simprr 772 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝑢 ∈ (Base‘𝐶))
12818, 122, 123, 20, 4, 124, 125, 25, 126, 127xpcid 18235 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩) = ⟨((Id‘𝑄)‘𝑓), ((Id‘𝐶)‘𝑢)⟩)
129128fveq2d 6909 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)‘⟨((Id‘𝑄)‘𝑓), ((Id‘𝐶)‘𝑢)⟩))
130 df-ov 7435 . . . . . . . . 9 (((Id‘𝑄)‘𝑓)(⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)((Id‘𝐶)‘𝑢)) = ((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)‘⟨((Id‘𝑄)‘𝑓), ((Id‘𝐶)‘𝑢)⟩)
131129, 130eqtr4di 2794 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = (((Id‘𝑄)‘𝑓)(⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)((Id‘𝐶)‘𝑢)))
1323adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → 𝐷 ∈ Cat)
133 eqid 2736 . . . . . . . . 9 (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩) = (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)
13420, 91, 124, 122, 126catidcl 17726 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝑄)‘𝑓) ∈ (𝑓(𝐶 Nat 𝐷)𝑓))
1354, 5, 125, 123, 127catidcl 17726 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝐶)‘𝑢) ∈ (𝑢(Hom ‘𝐶)𝑢))
1361, 123, 132, 4, 5, 6, 7, 126, 126, 127, 127, 133, 134, 135evlf2val 18265 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝑄)‘𝑓)(⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)((Id‘𝐶)‘𝑢)) = ((((Id‘𝑄)‘𝑓)‘𝑢)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑢)⟩(comp‘𝐷)((1st𝑓)‘𝑢))((𝑢(2nd𝑓)𝑢)‘((Id‘𝐶)‘𝑢))))
13730, 126, 32sylancr 587 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (1st𝑓)(𝐶 Func 𝐷)(2nd𝑓))
1384, 22, 137funcf1 17912 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (1st𝑓):(Base‘𝐶)⟶(Base‘𝐷))
139138, 127ffvelcdmd 7104 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((1st𝑓)‘𝑢) ∈ (Base‘𝐷))
14022, 24, 26, 132, 139catidcl 17726 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝐷)‘((1st𝑓)‘𝑢)) ∈ (((1st𝑓)‘𝑢)(Hom ‘𝐷)((1st𝑓)‘𝑢)))
14122, 24, 26, 132, 139, 6, 139, 140catlid 17727 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝐷)‘((1st𝑓)‘𝑢))(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑢)⟩(comp‘𝐷)((1st𝑓)‘𝑢))((Id‘𝐷)‘((1st𝑓)‘𝑢))) = ((Id‘𝐷)‘((1st𝑓)‘𝑢)))
14219, 124, 26, 126fucid 18020 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝑄)‘𝑓) = ((Id‘𝐷) ∘ (1st𝑓)))
143142fveq1d 6907 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝑄)‘𝑓)‘𝑢) = (((Id‘𝐷) ∘ (1st𝑓))‘𝑢))
144 fvco3 7007 . . . . . . . . . . . 12 (((1st𝑓):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑢 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ (1st𝑓))‘𝑢) = ((Id‘𝐷)‘((1st𝑓)‘𝑢)))
145138, 127, 144syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝐷) ∘ (1st𝑓))‘𝑢) = ((Id‘𝐷)‘((1st𝑓)‘𝑢)))
146143, 145eqtrd 2776 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (((Id‘𝑄)‘𝑓)‘𝑢) = ((Id‘𝐷)‘((1st𝑓)‘𝑢)))
1474, 125, 26, 137, 127funcid 17916 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((𝑢(2nd𝑓)𝑢)‘((Id‘𝐶)‘𝑢)) = ((Id‘𝐷)‘((1st𝑓)‘𝑢)))
148146, 147oveq12d 7450 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((((Id‘𝑄)‘𝑓)‘𝑢)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑢)⟩(comp‘𝐷)((1st𝑓)‘𝑢))((𝑢(2nd𝑓)𝑢)‘((Id‘𝐶)‘𝑢))) = (((Id‘𝐷)‘((1st𝑓)‘𝑢))(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑢)⟩(comp‘𝐷)((1st𝑓)‘𝑢))((Id‘𝐷)‘((1st𝑓)‘𝑢))))
1491, 123, 132, 4, 126, 127evlf1 18266 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → (𝑓(1st𝐸)𝑢) = ((1st𝑓)‘𝑢))
150149fveq2d 6909 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((Id‘𝐷)‘(𝑓(1st𝐸)𝑢)) = ((Id‘𝐷)‘((1st𝑓)‘𝑢)))
151141, 148, 1503eqtr4d 2786 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((((Id‘𝑄)‘𝑓)‘𝑢)(⟨((1st𝑓)‘𝑢), ((1st𝑓)‘𝑢)⟩(comp‘𝐷)((1st𝑓)‘𝑢))((𝑢(2nd𝑓)𝑢)‘((Id‘𝐶)‘𝑢))) = ((Id‘𝐷)‘(𝑓(1st𝐸)𝑢)))
152131, 136, 1513eqtrd 2780 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (𝐶 Func 𝐷) ∧ 𝑢 ∈ (Base‘𝐶))) → ((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((Id‘𝐷)‘(𝑓(1st𝐸)𝑢)))
153152ralrimivva 3201 . . . . . 6 (𝜑 → ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑢 ∈ (Base‘𝐶)((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((Id‘𝐷)‘(𝑓(1st𝐸)𝑢)))
154 id 22 . . . . . . . . . 10 (𝑥 = ⟨𝑓, 𝑢⟩ → 𝑥 = ⟨𝑓, 𝑢⟩)
155154, 154oveq12d 7450 . . . . . . . . 9 (𝑥 = ⟨𝑓, 𝑢⟩ → (𝑥(2nd𝐸)𝑥) = (⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩))
156 fveq2 6905 . . . . . . . . 9 (𝑥 = ⟨𝑓, 𝑢⟩ → ((Id‘(𝑄 ×c 𝐶))‘𝑥) = ((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩))
157155, 156fveq12d 6912 . . . . . . . 8 (𝑥 = ⟨𝑓, 𝑢⟩ → ((𝑥(2nd𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)))
158112fveq2d 6909 . . . . . . . 8 (𝑥 = ⟨𝑓, 𝑢⟩ → ((Id‘𝐷)‘((1st𝐸)‘𝑥)) = ((Id‘𝐷)‘(𝑓(1st𝐸)𝑢)))
159157, 158eqeq12d 2752 . . . . . . 7 (𝑥 = ⟨𝑓, 𝑢⟩ → (((𝑥(2nd𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((1st𝐸)‘𝑥)) ↔ ((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((Id‘𝐷)‘(𝑓(1st𝐸)𝑢))))
160159ralxp 5851 . . . . . 6 (∀𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))((𝑥(2nd𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((1st𝐸)‘𝑥)) ↔ ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑢 ∈ (Base‘𝐶)((⟨𝑓, 𝑢⟩(2nd𝐸)⟨𝑓, 𝑢⟩)‘((Id‘(𝑄 ×c 𝐶))‘⟨𝑓, 𝑢⟩)) = ((Id‘𝐷)‘(𝑓(1st𝐸)𝑢)))
161153, 160sylibr 234 . . . . 5 (𝜑 → ∀𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))((𝑥(2nd𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((1st𝐸)‘𝑥)))
162161r19.21bi 3250 . . . 4 ((𝜑𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) → ((𝑥(2nd𝐸)𝑥)‘((Id‘(𝑄 ×c 𝐶))‘𝑥)) = ((Id‘𝐷)‘((1st𝐸)‘𝑥)))
16323ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝐶 ∈ Cat)
16433ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝐷 ∈ Cat)
165 simp21 1206 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
166 1st2nd2 8054 . . . . . . . . 9 (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
167165, 166syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
168167, 165eqeltrrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
169 opelxp 5720 . . . . . . 7 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↔ ((1st𝑥) ∈ (𝐶 Func 𝐷) ∧ (2nd𝑥) ∈ (Base‘𝐶)))
170168, 169sylib 218 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st𝑥) ∈ (𝐶 Func 𝐷) ∧ (2nd𝑥) ∈ (Base‘𝐶)))
171 simp22 1207 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
172 1st2nd2 8054 . . . . . . . . 9 (𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
173171, 172syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
174173, 171eqeltrrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
175 opelxp 5720 . . . . . . 7 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↔ ((1st𝑦) ∈ (𝐶 Func 𝐷) ∧ (2nd𝑦) ∈ (Base‘𝐶)))
176174, 175sylib 218 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st𝑦) ∈ (𝐶 Func 𝐷) ∧ (2nd𝑦) ∈ (Base‘𝐶)))
177 simp23 1208 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
178 1st2nd2 8054 . . . . . . . . 9 (𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
179177, 178syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
180179, 177eqeltrrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)))
181 opelxp 5720 . . . . . . 7 (⟨(1st𝑧), (2nd𝑧)⟩ ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ↔ ((1st𝑧) ∈ (𝐶 Func 𝐷) ∧ (2nd𝑧) ∈ (Base‘𝐶)))
182180, 181sylib 218 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st𝑧) ∈ (𝐶 Func 𝐷) ∧ (2nd𝑧) ∈ (Base‘𝐶)))
183 simp3l 1201 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦))
18418, 21, 91, 5, 23, 165, 171xpchom 18226 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) = (((1st𝑥)(𝐶 Nat 𝐷)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))))
185183, 184eleqtrd 2842 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑓 ∈ (((1st𝑥)(𝐶 Nat 𝐷)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))))
186 1st2nd2 8054 . . . . . . . . 9 (𝑓 ∈ (((1st𝑥)(𝐶 Nat 𝐷)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
187185, 186syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑓 = ⟨(1st𝑓), (2nd𝑓)⟩)
188187, 185eqeltrrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st𝑓), (2nd𝑓)⟩ ∈ (((1st𝑥)(𝐶 Nat 𝐷)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))))
189 opelxp 5720 . . . . . . 7 (⟨(1st𝑓), (2nd𝑓)⟩ ∈ (((1st𝑥)(𝐶 Nat 𝐷)(1st𝑦)) × ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))) ↔ ((1st𝑓) ∈ ((1st𝑥)(𝐶 Nat 𝐷)(1st𝑦)) ∧ (2nd𝑓) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))))
190188, 189sylib 218 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st𝑓) ∈ ((1st𝑥)(𝐶 Nat 𝐷)(1st𝑦)) ∧ (2nd𝑓) ∈ ((2nd𝑥)(Hom ‘𝐶)(2nd𝑦))))
191 simp3r 1202 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))
19218, 21, 91, 5, 23, 171, 177xpchom 18226 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧) = (((1st𝑦)(𝐶 Nat 𝐷)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))))
193191, 192eleqtrd 2842 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑔 ∈ (((1st𝑦)(𝐶 Nat 𝐷)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))))
194 1st2nd2 8054 . . . . . . . . 9 (𝑔 ∈ (((1st𝑦)(𝐶 Nat 𝐷)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
195193, 194syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
196195, 193eqeltrrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨(1st𝑔), (2nd𝑔)⟩ ∈ (((1st𝑦)(𝐶 Nat 𝐷)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))))
197 opelxp 5720 . . . . . . 7 (⟨(1st𝑔), (2nd𝑔)⟩ ∈ (((1st𝑦)(𝐶 Nat 𝐷)(1st𝑧)) × ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))) ↔ ((1st𝑔) ∈ ((1st𝑦)(𝐶 Nat 𝐷)(1st𝑧)) ∧ (2nd𝑔) ∈ ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))))
198196, 197sylib 218 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st𝑔) ∈ ((1st𝑦)(𝐶 Nat 𝐷)(1st𝑧)) ∧ (2nd𝑔) ∈ ((2nd𝑦)(Hom ‘𝐶)(2nd𝑧))))
1991, 19, 163, 164, 7, 170, 176, 182, 190, 198evlfcllem 18267 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝐸)⟨(1st𝑧), (2nd𝑧)⟩)‘(⟨(1st𝑔), (2nd𝑔)⟩(⟨⟨(1st𝑥), (2nd𝑥)⟩, ⟨(1st𝑦), (2nd𝑦)⟩⟩(comp‘(𝑄 ×c 𝐶))⟨(1st𝑧), (2nd𝑧)⟩)⟨(1st𝑓), (2nd𝑓)⟩)) = (((⟨(1st𝑦), (2nd𝑦)⟩(2nd𝐸)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)(⟨((1st𝐸)‘⟨(1st𝑥), (2nd𝑥)⟩), ((1st𝐸)‘⟨(1st𝑦), (2nd𝑦)⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩))((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝐸)⟨(1st𝑦), (2nd𝑦)⟩)‘⟨(1st𝑓), (2nd𝑓)⟩)))
200167, 179oveq12d 7450 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑥(2nd𝐸)𝑧) = (⟨(1st𝑥), (2nd𝑥)⟩(2nd𝐸)⟨(1st𝑧), (2nd𝑧)⟩))
201167, 173opeq12d 4880 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨𝑥, 𝑦⟩ = ⟨⟨(1st𝑥), (2nd𝑥)⟩, ⟨(1st𝑦), (2nd𝑦)⟩⟩)
202201, 179oveq12d 7450 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (⟨𝑥, 𝑦⟩(comp‘(𝑄 ×c 𝐶))𝑧) = (⟨⟨(1st𝑥), (2nd𝑥)⟩, ⟨(1st𝑦), (2nd𝑦)⟩⟩(comp‘(𝑄 ×c 𝐶))⟨(1st𝑧), (2nd𝑧)⟩))
203202, 195, 187oveq123d 7453 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑄 ×c 𝐶))𝑧)𝑓) = (⟨(1st𝑔), (2nd𝑔)⟩(⟨⟨(1st𝑥), (2nd𝑥)⟩, ⟨(1st𝑦), (2nd𝑦)⟩⟩(comp‘(𝑄 ×c 𝐶))⟨(1st𝑧), (2nd𝑧)⟩)⟨(1st𝑓), (2nd𝑓)⟩))
204200, 203fveq12d 6912 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((𝑥(2nd𝐸)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑄 ×c 𝐶))𝑧)𝑓)) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝐸)⟨(1st𝑧), (2nd𝑧)⟩)‘(⟨(1st𝑔), (2nd𝑔)⟩(⟨⟨(1st𝑥), (2nd𝑥)⟩, ⟨(1st𝑦), (2nd𝑦)⟩⟩(comp‘(𝑄 ×c 𝐶))⟨(1st𝑧), (2nd𝑧)⟩)⟨(1st𝑓), (2nd𝑓)⟩)))
205167fveq2d 6909 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st𝐸)‘𝑥) = ((1st𝐸)‘⟨(1st𝑥), (2nd𝑥)⟩))
206173fveq2d 6909 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st𝐸)‘𝑦) = ((1st𝐸)‘⟨(1st𝑦), (2nd𝑦)⟩))
207205, 206opeq12d 4880 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ⟨((1st𝐸)‘𝑥), ((1st𝐸)‘𝑦)⟩ = ⟨((1st𝐸)‘⟨(1st𝑥), (2nd𝑥)⟩), ((1st𝐸)‘⟨(1st𝑦), (2nd𝑦)⟩)⟩)
208179fveq2d 6909 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩))
209207, 208oveq12d 7450 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (⟨((1st𝐸)‘𝑥), ((1st𝐸)‘𝑦)⟩(comp‘𝐷)((1st𝐸)‘𝑧)) = (⟨((1st𝐸)‘⟨(1st𝑥), (2nd𝑥)⟩), ((1st𝐸)‘⟨(1st𝑦), (2nd𝑦)⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩)))
210173, 179oveq12d 7450 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑦(2nd𝐸)𝑧) = (⟨(1st𝑦), (2nd𝑦)⟩(2nd𝐸)⟨(1st𝑧), (2nd𝑧)⟩))
211210, 195fveq12d 6912 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((𝑦(2nd𝐸)𝑧)‘𝑔) = ((⟨(1st𝑦), (2nd𝑦)⟩(2nd𝐸)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩))
212167, 173oveq12d 7450 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (𝑥(2nd𝐸)𝑦) = (⟨(1st𝑥), (2nd𝑥)⟩(2nd𝐸)⟨(1st𝑦), (2nd𝑦)⟩))
213212, 187fveq12d 6912 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((𝑥(2nd𝐸)𝑦)‘𝑓) = ((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝐸)⟨(1st𝑦), (2nd𝑦)⟩)‘⟨(1st𝑓), (2nd𝑓)⟩))
214209, 211, 213oveq123d 7453 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → (((𝑦(2nd𝐸)𝑧)‘𝑔)(⟨((1st𝐸)‘𝑥), ((1st𝐸)‘𝑦)⟩(comp‘𝐷)((1st𝐸)‘𝑧))((𝑥(2nd𝐸)𝑦)‘𝑓)) = (((⟨(1st𝑦), (2nd𝑦)⟩(2nd𝐸)⟨(1st𝑧), (2nd𝑧)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)(⟨((1st𝐸)‘⟨(1st𝑥), (2nd𝑥)⟩), ((1st𝐸)‘⟨(1st𝑦), (2nd𝑦)⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩))((⟨(1st𝑥), (2nd𝑥)⟩(2nd𝐸)⟨(1st𝑦), (2nd𝑦)⟩)‘⟨(1st𝑓), (2nd𝑓)⟩)))
215199, 204, 2143eqtr4d 2786 . . . 4 ((𝜑 ∧ (𝑥 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑦 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶)) ∧ 𝑧 ∈ ((𝐶 Func 𝐷) × (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝑄 ×c 𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝑄 ×c 𝐶))𝑧))) → ((𝑥(2nd𝐸)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘(𝑄 ×c 𝐶))𝑧)𝑓)) = (((𝑦(2nd𝐸)𝑧)‘𝑔)(⟨((1st𝐸)‘𝑥), ((1st𝐸)‘𝑦)⟩(comp‘𝐷)((1st𝐸)‘𝑧))((𝑥(2nd𝐸)𝑦)‘𝑓)))
21621, 22, 23, 24, 25, 26, 27, 6, 29, 3, 44, 55, 121, 162, 215isfuncd 17911 . . 3 (𝜑 → (1st𝐸)((𝑄 ×c 𝐶) Func 𝐷)(2nd𝐸))
217 df-br 5143 . . 3 ((1st𝐸)((𝑄 ×c 𝐶) Func 𝐷)(2nd𝐸) ↔ ⟨(1st𝐸), (2nd𝐸)⟩ ∈ ((𝑄 ×c 𝐶) Func 𝐷))
218216, 217sylib 218 . 2 (𝜑 → ⟨(1st𝐸), (2nd𝐸)⟩ ∈ ((𝑄 ×c 𝐶) Func 𝐷))
21917, 218eqeltrd 2840 1 (𝜑𝐸 ∈ ((𝑄 ×c 𝐶) Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  csb 3898  cop 4631   class class class wbr 5142   × cxp 5682  ccom 5688  Rel wrel 5689   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  1st c1st 8013  2nd c2nd 8014  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709   Func cfunc 17900   Nat cnat 17990   FuncCat cfuc 17991   ×c cxpc 18214   evalF cevlf 18255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-cat 17712  df-cid 17713  df-func 17904  df-nat 17992  df-fuc 17993  df-xpc 18218  df-evlf 18259
This theorem is referenced by:  uncfcl  18281  uncf1  18282  uncf2  18283  yonedalem1  18318
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