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Theorem curfuncf 17086
Description: Cancellation of curry with uncurry. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
uncfval.g 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
uncfval.c (𝜑𝐷 ∈ Cat)
uncfval.d (𝜑𝐸 ∈ Cat)
uncfval.f (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
Assertion
Ref Expression
curfuncf (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = 𝐺)

Proof of Theorem curfuncf
Dummy variables 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uncfval.g . . . . . . . . . 10 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2 uncfval.c . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
32ad2antrr 705 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
4 uncfval.d . . . . . . . . . . 11 (𝜑𝐸 ∈ Cat)
54ad2antrr 705 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
6 uncfval.f . . . . . . . . . . 11 (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
76ad2antrr 705 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
8 eqid 2771 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
9 eqid 2771 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
10 simplr 752 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
11 simpr 471 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
121, 3, 5, 7, 8, 9, 10, 11uncf1 17084 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(1st𝐹)𝑦) = ((1st ‘((1st𝐺)‘𝑥))‘𝑦))
1312mpteq2dva 4879 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ ((1st ‘((1st𝐺)‘𝑥))‘𝑦)))
14 eqid 2771 . . . . . . . . . 10 (Base‘𝐸) = (Base‘𝐸)
15 relfunc 16729 . . . . . . . . . . 11 Rel (𝐷 Func 𝐸)
16 eqid 2771 . . . . . . . . . . . . . 14 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
1716fucbas 16827 . . . . . . . . . . . . 13 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
18 relfunc 16729 . . . . . . . . . . . . . 14 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
19 1st2ndbr 7370 . . . . . . . . . . . . . 14 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
2018, 6, 19sylancr 575 . . . . . . . . . . . . 13 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
218, 17, 20funcf1 16733 . . . . . . . . . . . 12 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(𝐷 Func 𝐸))
2221ffvelrnda 6504 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
23 1st2ndbr 7370 . . . . . . . . . . 11 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
2415, 22, 23sylancr 575 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
259, 14, 24funcf1 16733 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
2625feqmptd 6393 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥)) = (𝑦 ∈ (Base‘𝐷) ↦ ((1st ‘((1st𝐺)‘𝑥))‘𝑦)))
2713, 26eqtr4d 2808 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)) = (1st ‘((1st𝐺)‘𝑥)))
282ad3antrrr 709 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
294ad3antrrr 709 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐸 ∈ Cat)
306ad3antrrr 709 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
31 simpllr 760 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑥 ∈ (Base‘𝐶))
32 simplrl 762 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ (Base‘𝐷))
33 eqid 2771 . . . . . . . . . . . . . 14 (Hom ‘𝐶) = (Hom ‘𝐶)
34 eqid 2771 . . . . . . . . . . . . . 14 (Hom ‘𝐷) = (Hom ‘𝐷)
35 simprr 756 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐷))
3635adantr 466 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ (Base‘𝐷))
37 eqid 2771 . . . . . . . . . . . . . . 15 (Id‘𝐶) = (Id‘𝐶)
38 funcrcl 16730 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)) → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
396, 38syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
4039simpld 482 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ Cat)
4140ad3antrrr 709 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
428, 33, 37, 41, 31catidcl 16550 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
43 simpr 471 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
441, 28, 29, 30, 8, 9, 31, 32, 33, 34, 31, 36, 42, 43uncf2 17085 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔) = ((((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))‘𝑧)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑦), ((1st ‘((1st𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑥))‘𝑧))((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)‘𝑔)))
45 eqid 2771 . . . . . . . . . . . . . . . . . 18 (Id‘(𝐷 FuncCat 𝐸)) = (Id‘(𝐷 FuncCat 𝐸))
4620ad3antrrr 709 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
478, 37, 45, 46, 31funcid 16737 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷 FuncCat 𝐸))‘((1st𝐺)‘𝑥)))
48 eqid 2771 . . . . . . . . . . . . . . . . . 18 (Id‘𝐸) = (Id‘𝐸)
4922ad2antrr 705 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
5016, 45, 48, 49fucid 16838 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘(𝐷 FuncCat 𝐸))‘((1st𝐺)‘𝑥)) = ((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥))))
5147, 50eqtrd 2805 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥))))
5251fveq1d 6335 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))‘𝑧) = (((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥)))‘𝑧))
5325ad2antrr 705 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
54 fvco3 6419 . . . . . . . . . . . . . . . 16 (((1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸) ∧ 𝑧 ∈ (Base‘𝐷)) → (((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥)))‘𝑧) = ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑧)))
5553, 36, 54syl2anc 573 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐸) ∘ (1st ‘((1st𝐺)‘𝑥)))‘𝑧) = ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑧)))
5652, 55eqtrd 2805 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))‘𝑧) = ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑧)))
5756oveq1d 6811 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))‘𝑧)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑦), ((1st ‘((1st𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑥))‘𝑧))((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)‘𝑔)) = (((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑧))(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑦), ((1st ‘((1st𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑥))‘𝑧))((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)‘𝑔)))
58 eqid 2771 . . . . . . . . . . . . . 14 (Hom ‘𝐸) = (Hom ‘𝐸)
5953, 32ffvelrnd 6505 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑦) ∈ (Base‘𝐸))
60 eqid 2771 . . . . . . . . . . . . . 14 (comp‘𝐸) = (comp‘𝐸)
6153, 36ffvelrnd 6505 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑧) ∈ (Base‘𝐸))
6224adantr 466 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
63 simprl 754 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
649, 34, 58, 62, 63, 35funcf2 16735 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st ‘((1st𝐺)‘𝑥))‘𝑦)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑥))‘𝑧)))
6564ffvelrnda 6504 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)‘𝑔) ∈ (((1st ‘((1st𝐺)‘𝑥))‘𝑦)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑥))‘𝑧)))
6614, 58, 48, 29, 59, 60, 61, 65catlid 16551 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑧))(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑦), ((1st ‘((1st𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑥))‘𝑧))((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)‘𝑔)) = ((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)‘𝑔))
6744, 57, 663eqtrd 2809 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔) = ((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)‘𝑔))
6867mpteq2dva 4879 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ ((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)‘𝑔)))
6964feqmptd 6393 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ ((𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)‘𝑔)))
7068, 69eqtr4d 2808 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧))
71703impb 1107 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧))
7271mpt2eq3dva 6870 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)))
739, 24funcfn2 16736 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (2nd ‘((1st𝐺)‘𝑥)) Fn ((Base‘𝐷) × (Base‘𝐷)))
74 fnov 6919 . . . . . . . . 9 ((2nd ‘((1st𝐺)‘𝑥)) Fn ((Base‘𝐷) × (Base‘𝐷)) ↔ (2nd ‘((1st𝐺)‘𝑥)) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)))
7573, 74sylib 208 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (2nd ‘((1st𝐺)‘𝑥)) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)))
7672, 75eqtr4d 2808 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔))) = (2nd ‘((1st𝐺)‘𝑥)))
7727, 76opeq12d 4548 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ⟨(1st ‘((1st𝐺)‘𝑥)), (2nd ‘((1st𝐺)‘𝑥))⟩)
78 1st2nd 7367 . . . . . . 7 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → ((1st𝐺)‘𝑥) = ⟨(1st ‘((1st𝐺)‘𝑥)), (2nd ‘((1st𝐺)‘𝑥))⟩)
7915, 22, 78sylancr 575 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) = ⟨(1st ‘((1st𝐺)‘𝑥)), (2nd ‘((1st𝐺)‘𝑥))⟩)
8077, 79eqtr4d 2808 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ((1st𝐺)‘𝑥))
8180mpteq2dva 4879 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐺)‘𝑥)))
8221feqmptd 6393 . . . 4 (𝜑 → (1st𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐺)‘𝑥)))
8381, 82eqtr4d 2808 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) = (1st𝐺))
842ad3antrrr 709 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
854ad3antrrr 709 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
866ad3antrrr 709 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
87 simprl 754 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
8887ad2antrr 705 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
89 simpr 471 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑧 ∈ (Base‘𝐷))
90 simprr 756 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
9190ad2antrr 705 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐶))
92 simplr 752 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
93 eqid 2771 . . . . . . . . . . . . 13 (Id‘𝐷) = (Id‘𝐷)
949, 34, 93, 84, 89catidcl 16550 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((Id‘𝐷)‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
951, 84, 85, 86, 8, 9, 88, 89, 33, 34, 91, 89, 92, 94uncf2 17085 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)) = ((((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑧), ((1st ‘((1st𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑦))‘𝑧))((𝑧(2nd ‘((1st𝐺)‘𝑥))𝑧)‘((Id‘𝐷)‘𝑧))))
9622adantrr 696 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
9796adantr 466 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
9815, 97, 23sylancr 575 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
9998adantr 466 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
1009, 93, 48, 99, 89funcid 16737 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((𝑧(2nd ‘((1st𝐺)‘𝑥))𝑧)‘((Id‘𝐷)‘𝑧)) = ((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑧)))
101100oveq2d 6812 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑧), ((1st ‘((1st𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑦))‘𝑧))((𝑧(2nd ‘((1st𝐺)‘𝑥))𝑧)‘((Id‘𝐷)‘𝑧))) = ((((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑧), ((1st ‘((1st𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑦))‘𝑧))((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑧))))
1029, 14, 98funcf1 16733 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
103102ffvelrnda 6504 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑧) ∈ (Base‘𝐸))
10421ffvelrnda 6504 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
105104adantrl 695 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
106105adantr 466 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
107 1st2ndbr 7370 . . . . . . . . . . . . . . 15 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st𝐺)‘𝑦))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑦)))
10815, 106, 107sylancr 575 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑦))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑦)))
1099, 14, 108funcf1 16733 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑦)):(Base‘𝐷)⟶(Base‘𝐸))
110109ffvelrnda 6504 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑦))‘𝑧) ∈ (Base‘𝐸))
111 eqid 2771 . . . . . . . . . . . . 13 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
11216, 111fuchom 16828 . . . . . . . . . . . . . . . 16 (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
11320ad3antrrr 709 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
1148, 33, 112, 113, 88, 91funcf2 16735 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
115114, 92ffvelrnd 6505 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((𝑥(2nd𝐺)𝑦)‘𝑔) ∈ (((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
116111, 115nat1st2nd 16818 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((𝑥(2nd𝐺)𝑦)‘𝑔) ∈ (⟨(1st ‘((1st𝐺)‘𝑥)), (2nd ‘((1st𝐺)‘𝑥))⟩(𝐷 Nat 𝐸)⟨(1st ‘((1st𝐺)‘𝑦)), (2nd ‘((1st𝐺)‘𝑦))⟩))
117111, 116, 9, 58, 89natcl 16820 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧) ∈ (((1st ‘((1st𝐺)‘𝑥))‘𝑧)(Hom ‘𝐸)((1st ‘((1st𝐺)‘𝑦))‘𝑧)))
11814, 58, 48, 85, 103, 60, 110, 117catrid 16552 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧)(⟨((1st ‘((1st𝐺)‘𝑥))‘𝑧), ((1st ‘((1st𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑦))‘𝑧))((Id‘𝐸)‘((1st ‘((1st𝐺)‘𝑥))‘𝑧))) = (((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧))
11995, 101, 1183eqtrd 2809 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)) = (((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧))
120119mpteq2dva 4879 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) = (𝑧 ∈ (Base‘𝐷) ↦ (((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧)))
12120adantr 466 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
1228, 33, 112, 121, 87, 90funcf2 16735 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
123122ffvelrnda 6504 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘𝑔) ∈ (((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
124111, 123nat1st2nd 16818 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘𝑔) ∈ (⟨(1st ‘((1st𝐺)‘𝑥)), (2nd ‘((1st𝐺)‘𝑥))⟩(𝐷 Nat 𝐸)⟨(1st ‘((1st𝐺)‘𝑦)), (2nd ‘((1st𝐺)‘𝑦))⟩))
125111, 124, 9natfn 16821 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘𝑔) Fn (Base‘𝐷))
126 dffn5 6385 . . . . . . . . . 10 (((𝑥(2nd𝐺)𝑦)‘𝑔) Fn (Base‘𝐷) ↔ ((𝑥(2nd𝐺)𝑦)‘𝑔) = (𝑧 ∈ (Base‘𝐷) ↦ (((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧)))
127125, 126sylib 208 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘𝑔) = (𝑧 ∈ (Base‘𝐷) ↦ (((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧)))
128120, 127eqtr4d 2808 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) = ((𝑥(2nd𝐺)𝑦)‘𝑔))
129128mpteq2dva 4879 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐺)𝑦)‘𝑔)))
130122feqmptd 6393 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐺)𝑦)‘𝑔)))
131129, 130eqtr4d 2808 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) = (𝑥(2nd𝐺)𝑦))
1321313impb 1107 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) = (𝑥(2nd𝐺)𝑦))
133132mpt2eq3dva 6870 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
1348, 20funcfn2 16736 . . . . 5 (𝜑 → (2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
135 fnov 6919 . . . . 5 ((2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
136134, 135sylib 208 . . . 4 (𝜑 → (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
137133, 136eqtr4d 2808 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) = (2nd𝐺))
13883, 137opeq12d 4548 . 2 (𝜑 → ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩ = ⟨(1st𝐺), (2nd𝐺)⟩)
139 eqid 2771 . . 3 (⟨𝐶, 𝐷⟩ curryF 𝐹) = (⟨𝐶, 𝐷⟩ curryF 𝐹)
1401, 2, 4, 6uncfcl 17083 . . 3 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
141139, 8, 40, 2, 140, 9, 34, 37, 33, 93curfval 17071 . 2 (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
142 1st2nd 7367 . . 3 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
14318, 6, 142sylancr 575 . 2 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
144138, 141, 1433eqtr4d 2815 1 (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cop 4323   class class class wbr 4787  cmpt 4864   × cxp 5248  ccom 5254  Rel wrel 5255   Fn wfn 6025  wf 6026  cfv 6030  (class class class)co 6796  cmpt2 6798  1st c1st 7317  2nd c2nd 7318  ⟨“cs3 13796  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532  Idccid 16533   Func cfunc 16721   Nat cnat 16808   FuncCat cfuc 16809   curryF ccurf 17058   uncurryF cuncf 17059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-7 11290  df-8 11291  df-9 11292  df-n0 11500  df-z 11585  df-dec 11701  df-uz 11894  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-s2 13802  df-s3 13803  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-hom 16174  df-cco 16175  df-cat 16536  df-cid 16537  df-func 16725  df-cofu 16727  df-nat 16810  df-fuc 16811  df-xpc 17020  df-1stf 17021  df-2ndf 17022  df-prf 17023  df-evlf 17061  df-curf 17062  df-uncf 17063
This theorem is referenced by: (None)
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