Theorem uncf2 18249

Description: Value of the uncurry functor on a morphism. (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)))
uncf1.a	⊢ 𝐴 = (Base‘𝐶)
uncf1.b	⊢ 𝐵 = (Base‘𝐷)
uncf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
uncf1.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
uncf2.h	⊢ 𝐻 = (Hom ‘𝐶)
uncf2.j	⊢ 𝐽 = (Hom ‘𝐷)
uncf2.z	⊢ (𝜑 → 𝑍 ∈ 𝐴)
uncf2.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)
uncf2.r	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑍))
uncf2.s	⊢ (𝜑 → 𝑆 ∈ (𝑌𝐽𝑊))

Assertion

Ref	Expression
uncf2	⊢ (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑍, 𝑊⟩)𝑆) = ((((𝑋(2nd ‘𝐺)𝑍)‘𝑅)‘𝑊)(⟨((1st ‘((1st ‘𝐺)‘𝑋))‘𝑌), ((1st ‘((1st ‘𝐺)‘𝑋))‘𝑊)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑍))‘𝑊))((𝑌(2nd ‘((1st ‘𝐺)‘𝑋))𝑊)‘𝑆)))

Proof of Theorem uncf2

Step	Hyp	Ref	Expression
1		uncfval.g	. . . . . . 7 ⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2		uncfval.c	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
3		uncfval.d	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Cat)
4		uncfval.f	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
5	1, 2, 3, 4	uncfval 18246	. . . . . 6 ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
6	5	fveq2d 6880	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) = (2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
7	6	oveqd 7422	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑍, 𝑊⟩) = (⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩))
8	7	oveqd 7422	. . 3 ⊢ (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑍, 𝑊⟩)𝑆) = (𝑅(⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)𝑆))
9		df-ov 7408	. . . 4 ⊢ (𝑅(⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)𝑆) = ((⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)
10		eqid 2735	. . . . . 6 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11		uncf1.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐶)
12		uncf1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
13	10, 11, 12	xpcbas 18190	. . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
14		eqid 2735	. . . . . 6 ⊢ ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
15		eqid 2735	. . . . . 6 ⊢ ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
16		funcrcl 17876	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)) → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
17	4, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
18	17	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
19		eqid 2735	. . . . . . . 8 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
20	10, 18, 2, 19	1stfcl 18209	. . . . . . 7 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
21	20, 4	cofucl 17901	. . . . . 6 ⊢ (𝜑 → (𝐺 ∘func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
22		eqid 2735	. . . . . . 7 ⊢ (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
23	10, 18, 2, 22	2ndfcl 18210	. . . . . 6 ⊢ (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
24	14, 15, 21, 23	prfcl 18215	. . . . 5 ⊢ (𝜑 → ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
25		eqid 2735	. . . . . 6 ⊢ (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
26		eqid 2735	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
27	25, 26, 2, 3	evlfcl 18234	. . . . 5 ⊢ (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
28		uncf1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
29		uncf1.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
30	28, 29	opelxpd 5693	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
31		uncf2.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
32		uncf2.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
33	31, 32	opelxpd 5693	. . . . 5 ⊢ (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
34		eqid 2735	. . . . 5 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
35		uncf2.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑍))
36		uncf2.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (𝑌𝐽𝑊))
37	35, 36	opelxpd 5693	. . . . . 6 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
38		uncf2.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
39		uncf2.j	. . . . . . 7 ⊢ 𝐽 = (Hom ‘𝐷)
40	10, 11, 12, 38, 39, 28, 29, 31, 32, 34	xpchom2 18198	. . . . . 6 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) = ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
41	37, 40	eleqtrrd 2837	. . . . 5 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))
42	13, 24, 27, 30, 33, 34, 41	cofu2 17899	. . . 4 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)))
43	9, 42	eqtrid 2782	. . 3 ⊢ (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)𝑆) = ((((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)))
44	8, 43	eqtrd 2770	. 2 ⊢ (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑍, 𝑊⟩)𝑆) = ((((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)))
45	14, 13, 34, 21, 23, 30	prf1 18212	. . . . . 6 ⊢ (𝜑 → ((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st ‘(𝐺 ∘func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩)
46	13, 20, 4, 30	cofu1 17897	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st ‘𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
47	10, 13, 34, 18, 2, 19, 30	1stf1 18204	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
48		op1stg 8000	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
49	28, 29, 48	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
50	47, 49	eqtrd 2770	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
51	50	fveq2d 6880	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st ‘𝐺)‘𝑋))
52	46, 51	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st ‘𝐺)‘𝑋))
53	10, 13, 34, 18, 2, 22, 30	2ndf1 18207	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
54		op2ndg 8001	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
55	28, 29, 54	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
56	53, 55	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
57	52, 56	opeq12d 4857	. . . . . 6 ⊢ (𝜑 → ⟨((1st ‘(𝐺 ∘func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st ‘𝐺)‘𝑋), 𝑌⟩)
58	45, 57	eqtrd 2770	. . . . 5 ⊢ (𝜑 → ((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st ‘𝐺)‘𝑋), 𝑌⟩)
59	14, 13, 34, 21, 23, 33	prf1 18212	. . . . . 6 ⊢ (𝜑 → ((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩) = ⟨((1st ‘(𝐺 ∘func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩)⟩)
60	13, 20, 4, 33	cofu1 17897	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st ‘𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)))
61	10, 13, 34, 18, 2, 19, 33	1stf1 18204	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = (1st ‘⟨𝑍, 𝑊⟩))
62		op1stg 8000	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ 𝐴 ∧ 𝑊 ∈ 𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
63	31, 32, 62	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
64	61, 63	eqtrd 2770	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑍)
65	64	fveq2d 6880	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)) = ((1st ‘𝐺)‘𝑍))
66	60, 65	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st ‘𝐺)‘𝑍))
67	10, 13, 34, 18, 2, 22, 33	2ndf1 18207	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = (2nd ‘⟨𝑍, 𝑊⟩))
68		op2ndg 8001	. . . . . . . . 9 ⊢ ((𝑍 ∈ 𝐴 ∧ 𝑊 ∈ 𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
69	31, 32, 68	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
70	67, 69	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑊)
71	66, 70	opeq12d 4857	. . . . . 6 ⊢ (𝜑 → ⟨((1st ‘(𝐺 ∘func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩)⟩ = ⟨((1st ‘𝐺)‘𝑍), 𝑊⟩)
72	59, 71	eqtrd 2770	. . . . 5 ⊢ (𝜑 → ((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩) = ⟨((1st ‘𝐺)‘𝑍), 𝑊⟩)
73	58, 72	oveq12d 7423	. . . 4 ⊢ (𝜑 → (((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩)) = (⟨((1st ‘𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st ‘𝐺)‘𝑍), 𝑊⟩))
74	14, 13, 34, 21, 23, 30, 33, 41	prf2 18214	. . . . 5 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ⟨((⟨𝑋, 𝑌⟩(2nd ‘(𝐺 ∘func (𝐶 1stF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩), ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)⟩)
75	13, 20, 4, 30, 33, 34, 41	cofu2 17899	. . . . . . 7 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐺 ∘func (𝐶 1stF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐺)((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)))
76	50, 64	oveq12d 7423	. . . . . . . 8 ⊢ (𝜑 → (((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐺)((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)) = (𝑋(2nd ‘𝐺)𝑍))
77	10, 13, 34, 18, 2, 19, 30, 33	1stf2 18205	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
78	77	fveq1d 6878	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
79	41	fvresd 6896	. . . . . . . . 9 ⊢ (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (1st ‘⟨𝑅, 𝑆⟩))
80		op1stg 8000	. . . . . . . . . 10 ⊢ ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
81	35, 36, 80	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
82	78, 79, 81	3eqtrd 2774	. . . . . . . 8 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑅)
83	76, 82	fveq12d 6883	. . . . . . 7 ⊢ (𝜑 → ((((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd ‘𝐺)((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)) = ((𝑋(2nd ‘𝐺)𝑍)‘𝑅))
84	75, 83	eqtrd 2770	. . . . . 6 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐺 ∘func (𝐶 1stF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((𝑋(2nd ‘𝐺)𝑍)‘𝑅))
85	10, 13, 34, 18, 2, 22, 30, 33	2ndf2 18208	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩) = (2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
86	85	fveq1d 6878	. . . . . . 7 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
87	41	fvresd 6896	. . . . . . 7 ⊢ (𝜑 → ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (2nd ‘⟨𝑅, 𝑆⟩))
88		op2ndg 8001	. . . . . . . 8 ⊢ ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
89	35, 36, 88	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
90	86, 87, 89	3eqtrd 2774	. . . . . 6 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑆)
91	84, 90	opeq12d 4857	. . . . 5 ⊢ (𝜑 → ⟨((⟨𝑋, 𝑌⟩(2nd ‘(𝐺 ∘func (𝐶 1stF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩), ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)⟩ = ⟨((𝑋(2nd ‘𝐺)𝑍)‘𝑅), 𝑆⟩)
92	74, 91	eqtrd 2770	. . . 4 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ⟨((𝑋(2nd ‘𝐺)𝑍)‘𝑅), 𝑆⟩)
93	73, 92	fveq12d 6883	. . 3 ⊢ (𝜑 → ((((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)) = ((⟨((1st ‘𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st ‘𝐺)‘𝑍), 𝑊⟩)‘⟨((𝑋(2nd ‘𝐺)𝑍)‘𝑅), 𝑆⟩))
94		df-ov 7408	. . 3 ⊢ (((𝑋(2nd ‘𝐺)𝑍)‘𝑅)(⟨((1st ‘𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st ‘𝐺)‘𝑍), 𝑊⟩)𝑆) = ((⟨((1st ‘𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st ‘𝐺)‘𝑍), 𝑊⟩)‘⟨((𝑋(2nd ‘𝐺)𝑍)‘𝑅), 𝑆⟩)
95	93, 94	eqtr4di 2788	. 2 ⊢ (𝜑 → ((((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)) = (((𝑋(2nd ‘𝐺)𝑍)‘𝑅)(⟨((1st ‘𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st ‘𝐺)‘𝑍), 𝑊⟩)𝑆))
96		eqid 2735	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
97		eqid 2735	. . 3 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
98	26	fucbas 17976	. . . . 5 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
99		relfunc 17875	. . . . . 6 ⊢ Rel (𝐶 Func (𝐷 FuncCat 𝐸))
100		1st2ndbr 8041	. . . . . 6 ⊢ ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd ‘𝐺))
101	99, 4, 100	sylancr 587	. . . . 5 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd ‘𝐺))
102	11, 98, 101	funcf1 17879	. . . 4 ⊢ (𝜑 → (1st ‘𝐺):𝐴⟶(𝐷 Func 𝐸))
103	102, 28	ffvelcdmd 7075	. . 3 ⊢ (𝜑 → ((1st ‘𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
104	102, 31	ffvelcdmd 7075	. . 3 ⊢ (𝜑 → ((1st ‘𝐺)‘𝑍) ∈ (𝐷 Func 𝐸))
105		eqid 2735	. . 3 ⊢ (⟨((1st ‘𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st ‘𝐺)‘𝑍), 𝑊⟩) = (⟨((1st ‘𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st ‘𝐺)‘𝑍), 𝑊⟩)
106	26, 97	fuchom 17977	. . . . 5 ⊢ (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
107	11, 38, 106, 101, 28, 31	funcf2 17881	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘𝐺)𝑍):(𝑋𝐻𝑍)⟶(((1st ‘𝐺)‘𝑋)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑍)))
108	107, 35	ffvelcdmd 7075	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘𝐺)𝑍)‘𝑅) ∈ (((1st ‘𝐺)‘𝑋)(𝐷 Nat 𝐸)((1st ‘𝐺)‘𝑍)))
109	25, 2, 3, 12, 39, 96, 97, 103, 104, 29, 32, 105, 108, 36	evlf2val 18231	. 2 ⊢ (𝜑 → (((𝑋(2nd ‘𝐺)𝑍)‘𝑅)(⟨((1st ‘𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st ‘𝐺)‘𝑍), 𝑊⟩)𝑆) = ((((𝑋(2nd ‘𝐺)𝑍)‘𝑅)‘𝑊)(⟨((1st ‘((1st ‘𝐺)‘𝑋))‘𝑌), ((1st ‘((1st ‘𝐺)‘𝑋))‘𝑊)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑍))‘𝑊))((𝑌(2nd ‘((1st ‘𝐺)‘𝑋))𝑊)‘𝑆)))
110	44, 95, 109	3eqtrd 2774	1 ⊢ (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑍, 𝑊⟩)𝑆) = ((((𝑋(2nd ‘𝐺)𝑍)‘𝑅)‘𝑊)(⟨((1st ‘((1st ‘𝐺)‘𝑋))‘𝑌), ((1st ‘((1st ‘𝐺)‘𝑋))‘𝑊)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑍))‘𝑊))((𝑌(2nd ‘((1st ‘𝐺)‘𝑋))𝑊)‘𝑆)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ⟨cop 4607   class class class wbr 5119   × cxp 5652   ↾ cres 5656  Rel wrel 5659  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  ⟨“cs3 14861  Basecbs 17228  Hom chom 17282  compcco 17283  Catccat 17676   Func cfunc 17867   ∘_func ccofu 17869   Nat cnat 17957   FuncCat cfuc 17958   ×_c cxpc 18180   1^st_F c1stf 18181   2^nd_F c2ndf 18182   ⟨,⟩_F cprf 18183   eval_F cevlf 18221   uncurry_F cuncf 18223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-fzo 13672  df-hash 14349  df-word 14532  df-concat 14589  df-s1 14614  df-s2 14867  df-s3 14868  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-hom 17295  df-cco 17296  df-cat 17680  df-cid 17681  df-func 17871  df-cofu 17873  df-nat 17959  df-fuc 17960  df-xpc 18184  df-1stf 18185  df-2ndf 18186  df-prf 18187  df-evlf 18225  df-uncf 18227

This theorem is referenced by:  curfuncf  18250  uncfcurf  18251