Theorem curfuncf 18193

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 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
4		uncfval.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Cat)
5	4	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
6		uncfval.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
7	6	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
8		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝐷)
10		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
11		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
12	1, 3, 5, 7, 8, 9, 10, 11	uncf1 18191	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(1st ‘𝐹)𝑦) = ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦))
13	12	mpteq2dva 5248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ ((1st ‘((1st ‘𝐺)‘𝑥))‘𝑦)))
14		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝐸) = (Base‘𝐸)
15		relfunc 17814	. . . . . . . . . . 11 ⊢ Rel (𝐷 Func 𝐸)
16		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
17	16	fucbas 17914	. . . . . . . . . . . . 13 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
18		relfunc 17814	. . . . . . . . . . . . . 14 ⊢ Rel (𝐶 Func (𝐷 FuncCat 𝐸))
19		1st2ndbr 8030	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd ‘𝐺))
21	8, 17, 20	funcf1 17818	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(𝐷 Func 𝐸))
22	21	ffvelcdmda 7086	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
23		1st2ndbr 8030	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st ‘𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑥)))
25	9, 14, 24	funcf1 17818	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘((1st ‘𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
26	25	feqmptd 6960	. . . . . . . 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 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat)
29	4	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐸 ∈ Cat)
30	6	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
31		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑥 ∈ (Base‘𝐶))
32		simplrl 775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ (Base‘𝐷))
33		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
34		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
35		simprr 771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐷))
36	35	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ (Base‘𝐷))
37		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (Id‘𝐶) = (Id‘𝐶)
38		funcrcl 17815	. . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ Cat)
41	40	ad3antrrr 728	. . . . . . . . . . . . . . 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 17628	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 18192	. . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (Id‘(𝐷 FuncCat 𝐸)) = (Id‘(𝐷 FuncCat 𝐸))
46	20	ad3antrrr 728	. . . . . . . . . . . . . . . . . 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 17822	. . . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (Id‘𝐸) = (Id‘𝐸)
49	22	ad2antrr 724	. . . . . . . . . . . . . . . . . 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 17926	. . . . . . . . . . . . . . . . 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 6893	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 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 6990	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
59	53, 32	ffvelcdmd 7087	. . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . 14 ⊢ (comp‘𝐸) = (comp‘𝐸)
61	53, 36	ffvelcdmd 7087	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (1st ‘((1st ‘𝐺)‘𝑥))(𝐷 Func 𝐸)(2nd ‘((1st ‘𝐺)‘𝑥)))
63		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
64	9, 34, 58, 62, 63, 35	funcf2 17820	. . . . . . . . . . . . . . 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 7086	. . . . . . . . . . . . . 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 17629	. . . . . . . . . . . . 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 5248	. . . . . . . . . . 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 6960	. . . . . . . . . . 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 7488	. . . . . . . 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 17821	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (2nd ‘((1st ‘𝐺)‘𝑥)) Fn ((Base‘𝐷) × (Base‘𝐷)))
74		fnov 7542	. . . . . . . . 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 217	. . . . . . . 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 4881	. . . . . 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 8027	. . . . . . 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 587	. . . . . 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 5248	. . . 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 6960	. . . 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 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
85	4	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐸 ∈ Cat)
86	6	ad3antrrr 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))
88	87	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶))
89		simpr 485	. . . . . . . . . . . 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))
91	90	ad2antrr 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 2732	. . . . . . . . . . . . 13 ⊢ (Id‘𝐷) = (Id‘𝐷)
94	9, 34, 93, 84, 89	catidcl 17628	. . . . . . . . . . . 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 18192	. . . . . . . . . . 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 715	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
97	96	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ‘𝐺)‘𝑥) ∈ (𝐷 Func 𝐸))
98	15, 97, 23	sylancr 587	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 17822	. . . . . . . . . . . 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 7427	. . . . . . . . . . 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 17818	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st ‘𝐺)‘𝑥)):(Base‘𝐷)⟶(Base‘𝐸))
103	102	ffvelcdmda 7086	. . . . . . . . . . . 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 7086	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
105	104	adantrl 714	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
106	105	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ‘𝐺)‘𝑦) ∈ (𝐷 Func 𝐸))
107		1st2ndbr 8030	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 17818	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((1st ‘𝐺)‘𝑦)):(Base‘𝐷)⟶(Base‘𝐸))
110	109	ffvelcdmda 7086	. . . . . . . . . . . 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 2732	. . . . . . . . . . . . 13 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
112	16, 111	fuchom 17915	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
113	20	ad3antrrr 728	. . . . . . . . . . . . . . . 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 17820	. . . . . . . . . . . . . . 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 7087	. . . . . . . . . . . . . 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 17904	. . . . . . . . . . . . 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 17906	. . . . . . . . . . . 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 17630	. . . . . . . . . . 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 5248	. . . . . . . . 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 481	. . . . . . . . . . . . . 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 17820	. . . . . . . . . . . . 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 7086	. . . . . . . . . . . 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 17904	. . . . . . . . . . 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 17907	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐺)𝑦)‘𝑔) Fn (Base‘𝐷))
126		dffn5 6950	. . . . . . . . . 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 217	. . . . . . . . 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 5248	. . . . . . 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 6960	. . . . . . 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 7488	. . . 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 17821	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
135		fnov 7542	. . . . 5 ⊢ ((2nd ‘𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐺)𝑦)))
136	134, 135	sylib 217	. . . 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 4881	. 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 2732	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF 𝐹) = (⟨𝐶, 𝐷⟩ curryF 𝐹)
140	1, 2, 4, 6	uncfcl 18190	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
141	139, 8, 40, 2, 140, 9, 34, 37, 33, 93	curfval 18178	. 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 8027	. . 3 ⊢ ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
143	18, 6, 142	sylancr 587	. 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 396   = wceq 1541   ∈ wcel 2106  ⟨cop 4634   class class class wbr 5148   ↦ cmpt 5231   × cxp 5674   ∘ ccom 5680  Rel wrel 5681   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7975  2^nd c2nd 7976  ⟨“cs3 14795  Basecbs 17146  Hom chom 17210  compcco 17211  Catccat 17610  Idccid 17611   Func cfunc 17806   Nat cnat 17894   FuncCat cfuc 17895   curry_F ccurf 18165   uncurry_F cuncf 18166

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-uz 12825  df-fz 13487  df-fzo 13630  df-hash 14293  df-word 14467  df-concat 14523  df-s1 14548  df-s2 14801  df-s3 14802  df-struct 17082  df-slot 17117  df-ndx 17129  df-base 17147  df-hom 17223  df-cco 17224  df-cat 17614  df-cid 17615  df-func 17810  df-cofu 17812  df-nat 17896  df-fuc 17897  df-xpc 18126  df-1stf 18127  df-2ndf 18128  df-prf 18129  df-evlf 18168  df-curf 18169  df-uncf 18170

This theorem is referenced by: (None)