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Theorem curfuncf 18132
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 725 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
4 uncfval.d . . . . . . . . . . 11 (𝜑𝐸 ∈ Cat)
54ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
6 uncfval.f . . . . . . . . . . 11 (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
76ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
8 eqid 2733 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
9 eqid 2733 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
10 simplr 768 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
11 simpr 486 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
121, 3, 5, 7, 8, 9, 10, 11uncf1 18130 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(1st𝐹)𝑦) = ((1st ‘((1st𝐺)‘𝑥))‘𝑦))
1312mpteq2dva 5206 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ ((1st ‘((1st𝐺)‘𝑥))‘𝑦)))
14 eqid 2733 . . . . . . . . . 10 (Base‘𝐸) = (Base‘𝐸)
15 relfunc 17753 . . . . . . . . . . 11 Rel (𝐷 Func 𝐸)
16 eqid 2733 . . . . . . . . . . . . . 14 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
1716fucbas 17853 . . . . . . . . . . . . 13 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
18 relfunc 17753 . . . . . . . . . . . . . 14 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
19 1st2ndbr 7975 . . . . . . . . . . . . . 14 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
2018, 6, 19sylancr 588 . . . . . . . . . . . . 13 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
218, 17, 20funcf1 17757 . . . . . . . . . . . 12 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(𝐷 Func 𝐸))
2221ffvelcdmda 7036 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
23 1st2ndbr 7975 . . . . . . . . . . 11 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
2415, 22, 23sylancr 588 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
259, 14, 24funcf1 17757 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
2625feqmptd 6911 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st𝐺)‘𝑥)) = (𝑦 ∈ (Base‘𝐷) ↦ ((1st ‘((1st𝐺)‘𝑥))‘𝑦)))
2713, 26eqtr4d 2776 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st𝐹)𝑦)) = (1st ‘((1st𝐺)‘𝑥)))
282ad3antrrr 729 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
294ad3antrrr 729 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐸 ∈ Cat)
306ad3antrrr 729 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
31 simpllr 775 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑥 ∈ (Base‘𝐶))
32 simplrl 776 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ (Base‘𝐷))
33 eqid 2733 . . . . . . . . . . . . . 14 (Hom ‘𝐶) = (Hom ‘𝐶)
34 eqid 2733 . . . . . . . . . . . . . 14 (Hom ‘𝐷) = (Hom ‘𝐷)
35 simprr 772 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐷))
3635adantr 482 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ (Base‘𝐷))
37 eqid 2733 . . . . . . . . . . . . . . 15 (Id‘𝐶) = (Id‘𝐶)
38 funcrcl 17754 . . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ Cat)
4140ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
428, 33, 37, 41, 31catidcl 17567 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
43 simpr 486 . . . . . . . . . . . . . 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 18131 . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . . 18 (Id‘(𝐷 FuncCat 𝐸)) = (Id‘(𝐷 FuncCat 𝐸))
4620ad3antrrr 729 . . . . . . . . . . . . . . . . . 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 17761 . . . . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . . 18 (Id‘𝐸) = (Id‘𝐸)
4922ad2antrr 725 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
5016, 45, 48, 49fucid 17865 . . . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . . . 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 6845 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
54 fvco3 6941 . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . 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 7373 . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . 14 (Hom ‘𝐸) = (Hom ‘𝐸)
5953, 32ffvelcdmd 7037 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑦) ∈ (Base‘𝐸))
60 eqid 2733 . . . . . . . . . . . . . 14 (comp‘𝐸) = (comp‘𝐸)
6153, 36ffvelcdmd 7037 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑧) ∈ (Base‘𝐸))
6224adantr 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
63 simprl 770 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
649, 34, 58, 62, 63, 35funcf2 17759 . . . . . . . . . . . . . . 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 7036 . . . . . . . . . . . . . 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 17568 . . . . . . . . . . . . 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 2777 . . . . . . . . . . . 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 5206 . . . . . . . . . . 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 6911 . . . . . . . . . . 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 2776 . . . . . . . . . 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 1116 . . . . . . . . 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 7435 . . . . . . . 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 17760 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶)) → (2nd ‘((1st𝐺)‘𝑥)) Fn ((Base‘𝐷) × (Base‘𝐷)))
74 fnov 7488 . . . . . . . . 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 2776 . . . . . . 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 4839 . . . . . 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 7972 . . . . . . 7 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → ((1st𝐺)‘𝑥) = ⟨(1st ‘((1st𝐺)‘𝑥)), (2nd ‘((1st𝐺)‘𝑥))⟩)
7915, 22, 78sylancr 588 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) = ⟨(1st ‘((1st𝐺)‘𝑥)), (2nd ‘((1st𝐺)‘𝑥))⟩)
8077, 79eqtr4d 2776 . . . . 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 5206 . . . 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 6911 . . . 4 (𝜑 → (1st𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐺)‘𝑥)))
8381, 82eqtr4d 2776 . . 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 729 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
854ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
866ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
87 simprl 770 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
8887ad2antrr 725 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
89 simpr 486 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑧 ∈ (Base‘𝐷))
90 simprr 772 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
9190ad2antrr 725 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐶))
92 simplr 768 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
93 eqid 2733 . . . . . . . . . . . . 13 (Id‘𝐷) = (Id‘𝐷)
949, 34, 93, 84, 89catidcl 17567 . . . . . . . . . . . 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 18131 . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
9796adantr 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
9815, 97, 23sylancr 588 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑥)))
9998adantr 482 . . . . . . . . . . . . 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 17761 . . . . . . . . . . . 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 7374 . . . . . . . . . . 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 17757 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
103102ffvelcdmda 7036 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑥))‘𝑧) ∈ (Base‘𝐸))
10421ffvelcdmda 7036 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
105104adantrl 715 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
106105adantr 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
107 1st2ndbr 7975 . . . . . . . . . . . . . . 15 ((Rel (𝐷 Func 𝐸) ∧ ((1st𝐺)‘𝑦) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st𝐺)‘𝑦))(𝐷 Func 𝐸)(2nd ‘((1st𝐺)‘𝑦)))
10815, 106, 107sylancr 588 . . . . . . . . . . . . . 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 17757 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st𝐺)‘𝑦)):(Base‘𝐷)⟶(Base‘𝐸))
110109ffvelcdmda 7036 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((1st ‘((1st𝐺)‘𝑦))‘𝑧) ∈ (Base‘𝐸))
111 eqid 2733 . . . . . . . . . . . . 13 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
11216, 111fuchom 17854 . . . . . . . . . . . . . . . 16 (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
11320ad3antrrr 729 . . . . . . . . . . . . . . . 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 17759 . . . . . . . . . . . . . . 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 7037 . . . . . . . . . . . . . 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 17843 . . . . . . . . . . . . 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 17845 . . . . . . . . . . . 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 17569 . . . . . . . . . . 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 2777 . . . . . . . . . 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 5206 . . . . . . . . 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 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
1228, 33, 112, 121, 87, 90funcf2 17759 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st𝐺)‘𝑦)))
123122ffvelcdmda 7036 . . . . . . . . . . . 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 17843 . . . . . . . . . . 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 17846 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘𝑔) Fn (Base‘𝐷))
126 dffn5 6902 . . . . . . . . . 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 2776 . . . . . . . 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 5206 . . . . . . 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 6911 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐺)𝑦)‘𝑔)))
131129, 130eqtr4d 2776 . . . . . 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 1116 . . . . 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 7435 . . . 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 17760 . . . . 5 (𝜑 → (2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
135 fnov 7488 . . . . 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 2776 . . 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 4839 . 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 2733 . . 3 (⟨𝐶, 𝐷⟩ curryF 𝐹) = (⟨𝐶, 𝐷⟩ curryF 𝐹)
1401, 2, 4, 6uncfcl 18129 . . 3 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
141139, 8, 40, 2, 140, 9, 34, 37, 33, 93curfval 18117 . 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 7972 . . 3 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
14318, 6, 142sylancr 588 . 2 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
144138, 141, 1433eqtr4d 2783 1 (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cop 4593   class class class wbr 5106  cmpt 5189   × cxp 5632  ccom 5638  Rel wrel 5639   Fn wfn 6492  wf 6493  cfv 6497  (class class class)co 7358  cmpo 7360  1st c1st 7920  2nd c2nd 7921  ⟨“cs3 14737  Basecbs 17088  Hom chom 17149  compcco 17150  Catccat 17549  Idccid 17550   Func cfunc 17745   Nat cnat 17833   FuncCat cfuc 17834   curryF ccurf 18104   uncurryF cuncf 18105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-fzo 13574  df-hash 14237  df-word 14409  df-concat 14465  df-s1 14490  df-s2 14743  df-s3 14744  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-func 17749  df-cofu 17751  df-nat 17835  df-fuc 17836  df-xpc 18065  df-1stf 18066  df-2ndf 18067  df-prf 18068  df-evlf 18107  df-curf 18108  df-uncf 18109
This theorem is referenced by: (None)
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