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Theorem curfuncf 18237
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 724 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
4 uncfval.d . . . . . . . . . . 11 (𝜑𝐸 ∈ Cat)
54ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
6 uncfval.f . . . . . . . . . . 11 (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
76ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
8 eqid 2728 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
9 eqid 2728 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
10 simplr 767 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
11 simpr 483 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
121, 3, 5, 7, 8, 9, 10, 11uncf1 18235 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(1st𝐹)𝑦) = ((1st ‘((1st𝐺)‘𝑥))‘𝑦))
1312mpteq2dva 5252 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ ((1st ‘((1st𝐺)‘𝑥))‘𝑦)))
14 eqid 2728 . . . . . . . . . 10 (Base‘𝐸) = (Base‘𝐸)
15 relfunc 17855 . . . . . . . . . . 11 Rel (𝐷 Func 𝐸)
16 eqid 2728 . . . . . . . . . . . . . 14 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
1716fucbas 17958 . . . . . . . . . . . . 13 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
18 relfunc 17855 . . . . . . . . . . . . . 14 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
19 1st2ndbr 8052 . . . . . . . . . . . . . 14 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
2018, 6, 19sylancr 585 . . . . . . . . . . . . 13 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
218, 17, 20funcf1 17859 . . . . . . . . . . . 12 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(𝐷 Func 𝐸))
2221ffvelcdmda 7099 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
23 1st2ndbr 8052 . . . . . . . . . . 11 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
2415, 22, 23sylancr 585 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
259, 14, 24funcf1 17859 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
2625feqmptd 6972 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥)) = (𝑦 ∈ (Base‘𝐷) ↦ ((1st ‘((1st𝐺)‘𝑥))‘𝑦)))
2713, 26eqtr4d 2771 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)) = (1st ‘((1st𝐺)‘𝑥)))
282ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
294ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐸 ∈ Cat)
306ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
31 simpllr 774 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑥 ∈ (Base‘𝐶))
32 simplrl 775 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ (Base‘𝐷))
33 eqid 2728 . . . . . . . . . . . . . 14 (Hom ‘𝐶) = (Hom ‘𝐶)
34 eqid 2728 . . . . . . . . . . . . . 14 (Hom ‘𝐷) = (Hom ‘𝐷)
35 simprr 771 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐷))
3635adantr 479 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ (Base‘𝐷))
37 eqid 2728 . . . . . . . . . . . . . . 15 (Id‘𝐶) = (Id‘𝐶)
38 funcrcl 17856 . . . . . . . . . . . . . . . . . 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 493 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ Cat)
4140ad3antrrr 728 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
428, 33, 37, 41, 31catidcl 17669 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
43 simpr 483 . . . . . . . . . . . . . 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 18236 . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . . . . . 18 (Id‘(𝐷 FuncCat 𝐸)) = (Id‘(𝐷 FuncCat 𝐸))
4620ad3antrrr 728 . . . . . . . . . . . . . . . . . 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 17863 . . . . . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . . . . . 18 (Id‘𝐸) = (Id‘𝐸)
4922ad2antrr 724 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
5016, 45, 48, 49fucid 17970 . . . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
54 fvco3 7002 . . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . 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 7441 . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . 14 (Hom ‘𝐸) = (Hom ‘𝐸)
5953, 32ffvelcdmd 7100 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑦) ∈ (Base‘𝐸))
60 eqid 2728 . . . . . . . . . . . . . 14 (comp‘𝐸) = (comp‘𝐸)
6153, 36ffvelcdmd 7100 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑧) ∈ (Base‘𝐸))
6224adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
63 simprl 769 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
649, 34, 58, 62, 63, 35funcf2 17861 . . . . . . . . . . . . . . 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)))
6564ffvelcdmda 7099 . . . . . . . . . . . . . 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 17670 . . . . . . . . . . . . 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 2772 . . . . . . . . . . . 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 5252 . . . . . . . . . . 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 6972 . . . . . . . . . . 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 2771 . . . . . . . . . 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 1112 . . . . . . . . 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))
7271mpoeq3dva 7503 . . . . . . . 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 17862 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (2nd ‘((1st𝐺)‘𝑥)) Fn ((Base‘𝐷) × (Base‘𝐷)))
74 fnov 7558 . . . . . . . . 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 217 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (2nd ‘((1st𝐺)‘𝑥)) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑦(2nd ‘((1st𝐺)‘𝑥))𝑧)))
7672, 75eqtr4d 2771 . . . . . . 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 4886 . . . . . 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 8049 . . . . . . 7 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → ((1st𝐺)‘𝑥) = ⟨(1st ‘((1st𝐺)‘𝑥)), (2nd ‘((1st𝐺)‘𝑥))⟩)
7915, 22, 78sylancr 585 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) = ⟨(1st ‘((1st𝐺)‘𝑥)), (2nd ‘((1st𝐺)‘𝑥))⟩)
8077, 79eqtr4d 2771 . . . . 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 5252 . . . 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 6972 . . . 4 (𝜑 → (1st𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐺)‘𝑥)))
8381, 82eqtr4d 2771 . . 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 728 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
854ad3antrrr 728 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
866ad3antrrr 728 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
87 simprl 769 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
8887ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
89 simpr 483 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑧 ∈ (Base‘𝐷))
90 simprr 771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
9190ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐶))
92 simplr 767 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
93 eqid 2728 . . . . . . . . . . . . 13 (Id‘𝐷) = (Id‘𝐷)
949, 34, 93, 84, 89catidcl 17669 . . . . . . . . . . . 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 18236 . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
9796adantr 479 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
9815, 97, 23sylancr 585 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
9998adantr 479 . . . . . . . . . . . . 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 17863 . . . . . . . . . . . 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 7442 . . . . . . . . . . 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 17859 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
103102ffvelcdmda 7099 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑧) ∈ (Base‘𝐸))
10421ffvelcdmda 7099 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
105104adantrl 714 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
106105adantr 479 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
107 1st2ndbr 8052 . . . . . . . . . . . . . . 15 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st𝐺)‘𝑦))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑦)))
10815, 106, 107sylancr 585 . . . . . . . . . . . . . 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 17859 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑦)):(Base‘𝐷)⟶(Base‘𝐸))
110109ffvelcdmda 7099 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑦))‘𝑧) ∈ (Base‘𝐸))
111 eqid 2728 . . . . . . . . . . . . 13 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
11216, 111fuchom 17959 . . . . . . . . . . . . . . . 16 (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
11320ad3antrrr 728 . . . . . . . . . . . . . . . 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 17861 . . . . . . . . . . . . . . 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, 92ffvelcdmd 7100 . . . . . . . . . . . . . 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 17948 . . . . . . . . . . . . 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 17950 . . . . . . . . . . . 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 17671 . . . . . . . . . . 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 2772 . . . . . . . . . 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 5252 . . . . . . . . 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 479 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
1228, 33, 112, 121, 87, 90funcf2 17861 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
123122ffvelcdmda 7099 . . . . . . . . . . . 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 17948 . . . . . . . . . . 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 17951 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘𝑔) Fn (Base‘𝐷))
126 dffn5 6962 . . . . . . . . . 10 (((𝑥(2nd𝐺)𝑦)‘𝑔) Fn (Base‘𝐷) ↔ ((𝑥(2nd𝐺)𝑦)‘𝑔) = (𝑧 ∈ (Base‘𝐷) ↦ (((𝑥(2nd𝐺)𝑦)‘𝑔)‘𝑧)))
127125, 126sylib 217 . . . . . . . . 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 2771 . . . . . . . 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 5252 . . . . . . 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 6972 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐺)𝑦)‘𝑔)))
131129, 130eqtr4d 2771 . . . . . 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 1112 . . . . 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))
133132mpoeq3dva 7503 . . . 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 17862 . . . . 5 (𝜑 → (2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
135 fnov 7558 . . . . 5 ((2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
136134, 135sylib 217 . . . 4 (𝜑 → (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
137133, 136eqtr4d 2771 . . 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 4886 . 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 2728 . . 3 (⟨𝐶, 𝐷⟩ curryF 𝐹) = (⟨𝐶, 𝐷⟩ curryF 𝐹)
1401, 2, 4, 6uncfcl 18234 . . 3 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
141139, 8, 40, 2, 140, 9, 34, 37, 33, 93curfval 18222 . 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 8049 . . 3 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
14318, 6, 142sylancr 585 . 2 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
144138, 141, 1433eqtr4d 2778 1 (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4638   class class class wbr 5152  cmpt 5235   × cxp 5680  ccom 5686  Rel wrel 5687   Fn wfn 6548  wf 6549  cfv 6553  (class class class)co 7426  cmpo 7428  1st c1st 7997  2nd c2nd 7998  ⟨“cs3 14833  Basecbs 17187  Hom chom 17251  compcco 17252  Catccat 17651  Idccid 17652   Func cfunc 17847   Nat cnat 17938   FuncCat cfuc 17939   curryF ccurf 18209   uncurryF cuncf 18210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-fzo 13668  df-hash 14330  df-word 14505  df-concat 14561  df-s1 14586  df-s2 14839  df-s3 14840  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17188  df-hom 17264  df-cco 17265  df-cat 17655  df-cid 17656  df-func 17851  df-cofu 17853  df-nat 17940  df-fuc 17941  df-xpc 18170  df-1stf 18171  df-2ndf 18172  df-prf 18173  df-evlf 18212  df-curf 18213  df-uncf 18214
This theorem is referenced by: (None)
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