Theorem curfuncf 18195

Description: Cancellation of curry with uncurry. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
uncfval.g	⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
uncfval.c	⊢ (𝜑 → 𝐷 ∈ Cat)
uncfval.d	⊢ (𝜑 → 𝐸 ∈ Cat)
uncfval.f	⊢ (𝜑 → 𝐺 ∈ (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.

Step	Hyp	Ref	Expression
1		uncfval.g	. . . . . . . . . 10 ⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2		uncfval.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ Cat)
3	2	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
4		uncfval.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Cat)
5	4	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
6		uncfval.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
7	6	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
8		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
10		simplr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
11		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
12	1, 3, 5, 7, 8, 9, 10, 11	uncf1 18193	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(1st ‘𝐹)𝑦) = ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦))
13	12	mpteq2dva 5179	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦)))
14		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝐸) = (Base‘𝐸)
15		relfunc 17820	. . . . . . . . . . 11 ⊢ Rel (𝐷 Func 𝐸)
16		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
17	16	fucbas 17921	. . . . . . . . . . . . 13 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
18		relfunc 17820	. . . . . . . . . . . . . 14 ⊢ Rel (𝐶 Func (𝐷 FuncCat 𝐸))
19		1st2ndbr 7988	. . . . . . . . . . . . . 14 ⊢ ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd ‘𝐺))
20	18, 6, 19	sylancr 588	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd ‘𝐺))
21	8, 17, 20	funcf1 17824	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(𝐷 Func 𝐸))
22	21	ffvelcdmda 7030	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
23		1st2ndbr 7988	. . . . . . . . . . 11 ⊢ ((Rel (𝐷 Func 𝐸) ∧ ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st ‘𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑥)))
24	15, 22, 23	sylancr 588	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st ‘𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑥)))
25	9, 14, 24	funcf1 17824	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st ‘𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
26	25	feqmptd 6902	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st ‘𝐺)‘𝑥)) = (𝑦 ∈ (Base‘𝐷) ↦ ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦)))
27	13, 26	eqtr4d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)) = (1st ‘((1st ‘𝐺)‘𝑥)))
28	2	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
29	4	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐸 ∈ Cat)
30	6	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
31		simpllr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑥 ∈ (Base‘𝐶))
32		simplrl 777	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ (Base‘𝐷))
33		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
34		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
35		simprr 773	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐷))
36	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ (Base‘𝐷))
37		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Id‘𝐶) = (Id‘𝐶)
38		funcrcl 17821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)) → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
39	6, 38	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
40	39	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ Cat)
41	40	ad3antrrr 731	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat)
42	8, 33, 37, 41, 31	catidcl 17639	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
43		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
44	1, 28, 29, 30, 8, 9, 31, 32, 33, 34, 31, 36, 42, 43	uncf2 18194	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔) = ((((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥))‘𝑧)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧))((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘𝑔)))
45		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (Id‘(𝐷 FuncCat 𝐸)) = (Id‘(𝐷 FuncCat 𝐸))
46	20	ad3antrrr 731	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd ‘𝐺))
47	8, 37, 45, 46, 31	funcid 17828	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷 FuncCat 𝐸))‘((1st ‘𝐺)‘𝑥)))
48		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (Id‘𝐸) = (Id‘𝐸)
49	22	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
50	16, 45, 48, 49	fucid 17932	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘(𝐷 FuncCat 𝐸))‘((1st ‘𝐺)‘𝑥)) = ((Id‘𝐸) ∘ (1st ‘((1st ‘𝐺)‘𝑥))))
51	47, 50	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸) ∘ (1st ‘((1st ‘𝐺)‘𝑥))))
52	51	fveq1d 6836	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥))‘𝑧) = (((Id‘𝐸) ∘ (1st ‘((1st ‘𝐺)‘𝑥)))‘𝑧))
53	25	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘((1st ‘𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
54		fvco3 6933	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘((1st ‘𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸) ∧ 𝑧 ∈ (Base‘𝐷)) → (((Id‘𝐸) ∘ (1st ‘((1st ‘𝐺)‘𝑥)))‘𝑧) = ((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)))
55	53, 36, 54	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐸) ∘ (1st ‘((1st ‘𝐺)‘𝑥)))‘𝑧) = ((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)))
56	52, 55	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥))‘𝑧) = ((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)))
57	56	oveq1d 7375	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((((𝑥(2nd ‘𝐺)𝑥)‘((Id‘𝐶)‘𝑥))‘𝑧)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧))((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘𝑔)) = (((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧))(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧))((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘𝑔)))
58		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
59	53, 32	ffvelcdmd 7031	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦) ∈ (Base‘𝐸))
60		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (comp‘𝐸) = (comp‘𝐸)
61	53, 36	ffvelcdmd 7031	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧) ∈ (Base‘𝐸))
62	24	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (1st ‘((1st ‘𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑥)))
63		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
64	9, 34, 58, 62, 63, 35	funcf2 17826	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦)(Hom ‘𝐸)((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)))
65	64	ffvelcdmda 7030	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘𝑔) ∈ (((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦)(Hom ‘𝐸)((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)))
66	14, 58, 48, 29, 59, 60, 61, 65	catlid 17640	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧))(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧))((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘𝑔)) = ((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘𝑔))
67	44, 57, 66	3eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔) = ((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘𝑔))
68	67	mpteq2dva 5179	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ ((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘𝑔)))
69	64	feqmptd 6902	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ ((𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘𝑔)))
70	68, 69	eqtr4d 2775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧))
71	70	3impb 1115	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧))
72	71	mpoeq3dva 7437	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)))
73	9, 24	funcfn2 17827	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (2nd ‘((1st ‘𝐺)‘𝑥)) Fn ((Base‘𝐷) × (Base‘𝐷)))
74		fnov 7491	. . . . . . . . 9 ⊢ ((2nd ‘((1st ‘𝐺)‘𝑥)) Fn ((Base‘𝐷) × (Base‘𝐷)) ↔ (2nd ‘((1st ‘𝐺)‘𝑥)) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)))
75	73, 74	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (2nd ‘((1st ‘𝐺)‘𝑥)) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑦(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)))
76	72, 75	eqtr4d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔))) = (2nd ‘((1st ‘𝐺)‘𝑥)))
77	27, 76	opeq12d 4825	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ⟨(1st ‘((1st ‘𝐺)‘𝑥)), (2nd ‘((1st ‘𝐺)‘𝑥))⟩)
78		1st2nd 7985	. . . . . . 7 ⊢ ((Rel (𝐷 Func 𝐸) ∧ ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸)) → ((1st ‘𝐺)‘𝑥) = ⟨(1st ‘((1st ‘𝐺)‘𝑥)), (2nd ‘((1st ‘𝐺)‘𝑥))⟩)
79	15, 22, 78	sylancr 588	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) = ⟨(1st ‘((1st ‘𝐺)‘𝑥)), (2nd ‘((1st ‘𝐺)‘𝑥))⟩)
80	77, 79	eqtr4d 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ((1st ‘𝐺)‘𝑥))
81	80	mpteq2dva 5179	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝐺)‘𝑥)))
82	21	feqmptd 6902	. . . 4 ⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝐺)‘𝑥)))
83	81, 82	eqtr4d 2775	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) = (1st ‘𝐺))
84	2	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
85	4	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
86	6	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
87		simprl 771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
88	87	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
89		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑧 ∈ (Base‘𝐷))
90		simprr 773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
91	90	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐶))
92		simplr 769	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
93		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Id‘𝐷) = (Id‘𝐷)
94	9, 34, 93, 84, 89	catidcl 17639	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((Id‘𝐷)‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧))
95	1, 84, 85, 86, 8, 9, 88, 89, 33, 34, 91, 89, 92, 94	uncf2 18194	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)) = ((((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑦))‘𝑧))((𝑧(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘((Id‘𝐷)‘𝑧))))
96	22	adantrr 718	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
97	96	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
98	15, 97, 23	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st ‘𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑥)))
99	98	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (1st ‘((1st ‘𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑥)))
100	9, 93, 48, 99, 89	funcid 17828	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((𝑧(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘((Id‘𝐷)‘𝑧)) = ((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)))
101	100	oveq2d 7376	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑦))‘𝑧))((𝑧(2nd ‘((1st ‘𝐺)‘𝑥))𝑧)‘((Id‘𝐷)‘𝑧))) = ((((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑦))‘𝑧))((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧))))
102	9, 14, 98	funcf1 17824	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st ‘𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
103	102	ffvelcdmda 7030	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧) ∈ (Base‘𝐸))
104	21	ffvelcdmda 7030	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
105	104	adantrl 717	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
106	105	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ‘𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
107		1st2ndbr 7988	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝐷 Func 𝐸) ∧ ((1st ‘𝐺)‘𝑦) ∈ (𝐷 Func 𝐸)) → (1st ‘((1st ‘𝐺)‘𝑦))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑦)))
108	15, 106, 107	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st ‘𝐺)‘𝑦))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑦)))
109	9, 14, 108	funcf1 17824	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st ‘𝐺)‘𝑦)):(Base‘𝐷)⟶(Base‘𝐸))
110	109	ffvelcdmda 7030	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐺)‘𝑦))‘𝑧) ∈ (Base‘𝐸))
111		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
112	16, 111	fuchom 17922	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
113	20	ad3antrrr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (1st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd ‘𝐺))
114	8, 33, 112, 113, 88, 91	funcf2 17826	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑥(2nd ‘𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑦)))
115	114, 92	ffvelcdmd 7031	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) ∈ (((1st ‘𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑦)))
116	111, 115	nat1st2nd 17912	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) ∈ (⟨(1st ‘((1st ‘𝐺)‘𝑥)), (2nd ‘((1st ‘𝐺)‘𝑥))⟩(𝐷 Nat 𝐸)⟨(1st ‘((1st ‘𝐺)‘𝑦)), (2nd ‘((1st ‘𝐺)‘𝑦))⟩))
117	111, 116, 9, 58, 89	natcl 17914	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧) ∈ (((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)(Hom ‘𝐸)((1st ‘((1st ‘𝐺)‘𝑦))‘𝑧)))
118	14, 58, 48, 85, 103, 60, 110, 117	catrid 17641	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧)(⟨((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧), ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑦))‘𝑧))((Id‘𝐸)‘((1st ‘((1st ‘𝐺)‘𝑥))‘𝑧))) = (((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧))
119	95, 101, 118	3eqtrd 2776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)) = (((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧))
120	119	mpteq2dva 5179	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) = (𝑧 ∈ (Base‘𝐷) ↦ (((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧)))
121	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd ‘𝐺))
122	8, 33, 112, 121, 87, 90	funcf2 17826	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑦)))
123	122	ffvelcdmda 7030	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) ∈ (((1st ‘𝐺)‘𝑥)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑦)))
124	111, 123	nat1st2nd 17912	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) ∈ (⟨(1st ‘((1st ‘𝐺)‘𝑥)), (2nd ‘((1st ‘𝐺)‘𝑥))⟩(𝐷 Nat 𝐸)⟨(1st ‘((1st ‘𝐺)‘𝑦)), (2nd ‘((1st ‘𝐺)‘𝑦))⟩))
125	111, 124, 9	natfn 17915	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) Fn (Base‘𝐷))
126		dffn5 6892	. . . . . . . . . 10 ⊢ (((𝑥(2nd ‘𝐺)𝑦)‘𝑔) Fn (Base‘𝐷) ↔ ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) = (𝑧 ∈ (Base‘𝐷) ↦ (((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧)))
127	125, 126	sylib 218	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) = (𝑧 ∈ (Base‘𝐷) ↦ (((𝑥(2nd ‘𝐺)𝑦)‘𝑔)‘𝑧)))
128	120, 127	eqtr4d 2775	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))) = ((𝑥(2nd ‘𝐺)𝑦)‘𝑔))
129	128	mpteq2dva 5179	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘𝐺)𝑦)‘𝑔)))
130	122	feqmptd 6902	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐺)𝑦) = (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘𝐺)𝑦)‘𝑔)))
131	129, 130	eqtr4d 2775	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) = (𝑥(2nd ‘𝐺)𝑦))
132	131	3impb 1115	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))) = (𝑥(2nd ‘𝐺)𝑦))
133	132	mpoeq3dva 7437	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐺)𝑦)))
134	8, 20	funcfn2 17827	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
135		fnov 7491	. . . . 5 ⊢ ((2nd ‘𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐺)𝑦)))
136	134, 135	sylib 218	. . . 4 ⊢ (𝜑 → (2nd ‘𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐺)𝑦)))
137	133, 136	eqtr4d 2775	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))) = (2nd ‘𝐺))
138	83, 137	opeq12d 4825	. 2 ⊢ (𝜑 → ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩ = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
139		eqid 2737	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF 𝐹) = (⟨𝐶, 𝐷⟩ curryF 𝐹)
140	1, 2, 4, 6	uncfcl 18192	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
141	139, 8, 40, 2, 140, 9, 34, 37, 33, 93	curfval 18180	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
142		1st2nd 7985	. . 3 ⊢ ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
143	18, 6, 142	sylancr 588	. 2 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
144	138, 141, 143	3eqtr4d 2782	1 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = 𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086   ↦ cmpt 5167   × cxp 5622   ∘ ccom 5628  Rel wrel 5629   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  1^st c1st 7933  2^nd c2nd 7934  ⟨“cs3 14795  Basecbs 17170  Hom chom 17222  compcco 17223  Catccat 17621  Idccid 17622   Func cfunc 17812   Nat cnat 17902   FuncCat cfuc 17903   curry_F ccurf 18167   uncurry_F cuncf 18168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-fzo 13600  df-hash 14284  df-word 14467  df-concat 14524  df-s1 14550  df-s2 14801  df-s3 14802  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-func 17816  df-cofu 17818  df-nat 17904  df-fuc 17905  df-xpc 18129  df-1stf 18130  df-2ndf 18131  df-prf 18132  df-evlf 18170  df-curf 18171  df-uncf 18172

This theorem is referenced by: (None)