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Theorem uncf2 16924
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 (𝜑𝐷 ∈ Cat)
uncfval.d (𝜑𝐸 ∈ Cat)
uncfval.f (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
uncf1.a 𝐴 = (Base‘𝐶)
uncf1.b 𝐵 = (Base‘𝐷)
uncf1.x (𝜑𝑋𝐴)
uncf1.y (𝜑𝑌𝐵)
uncf2.h 𝐻 = (Hom ‘𝐶)
uncf2.j 𝐽 = (Hom ‘𝐷)
uncf2.z (𝜑𝑍𝐴)
uncf2.w (𝜑𝑊𝐵)
uncf2.r (𝜑𝑅 ∈ (𝑋𝐻𝑍))
uncf2.s (𝜑𝑆 ∈ (𝑌𝐽𝑊))
Assertion
Ref Expression
uncf2 (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑍, 𝑊⟩)𝑆) = ((((𝑋(2nd𝐺)𝑍)‘𝑅)‘𝑊)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑌), ((1st ‘((1st𝐺)‘𝑋))‘𝑊)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑍))‘𝑊))((𝑌(2nd ‘((1st𝐺)‘𝑋))𝑊)‘𝑆)))

Proof of Theorem uncf2
StepHypRef Expression
1 uncfval.g . . . . . . 7 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2 uncfval.c . . . . . . 7 (𝜑𝐷 ∈ Cat)
3 uncfval.d . . . . . . 7 (𝜑𝐸 ∈ Cat)
4 uncfval.f . . . . . . 7 (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
51, 2, 3, 4uncfval 16921 . . . . . 6 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6233 . . . . 5 (𝜑 → (2nd𝐹) = (2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 6707 . . . 4 (𝜑 → (⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑍, 𝑊⟩) = (⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩))
87oveqd 6707 . . 3 (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑍, 𝑊⟩)𝑆) = (𝑅(⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)𝑆))
9 df-ov 6693 . . . 4 (𝑅(⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)𝑆) = ((⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)
10 eqid 2651 . . . . . 6 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11 uncf1.a . . . . . 6 𝐴 = (Base‘𝐶)
12 uncf1.b . . . . . 6 𝐵 = (Base‘𝐷)
1310, 11, 12xpcbas 16865 . . . . 5 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
14 eqid 2651 . . . . . 6 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
15 eqid 2651 . . . . . 6 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
16 funcrcl 16570 . . . . . . . . . 10 (𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)) → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
174, 16syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
1817simpld 474 . . . . . . . 8 (𝜑𝐶 ∈ Cat)
19 eqid 2651 . . . . . . . 8 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
2010, 18, 2, 191stfcl 16884 . . . . . . 7 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2120, 4cofucl 16595 . . . . . 6 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
22 eqid 2651 . . . . . . 7 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
2310, 18, 2, 222ndfcl 16885 . . . . . 6 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2414, 15, 21, 23prfcl 16890 . . . . 5 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
25 eqid 2651 . . . . . 6 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
26 eqid 2651 . . . . . 6 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2725, 26, 2, 3evlfcl 16909 . . . . 5 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
28 uncf1.x . . . . . 6 (𝜑𝑋𝐴)
29 uncf1.y . . . . . 6 (𝜑𝑌𝐵)
30 opelxpi 5182 . . . . . 6 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3128, 29, 30syl2anc 694 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
32 uncf2.z . . . . . 6 (𝜑𝑍𝐴)
33 uncf2.w . . . . . 6 (𝜑𝑊𝐵)
34 opelxpi 5182 . . . . . 6 ((𝑍𝐴𝑊𝐵) → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
3532, 33, 34syl2anc 694 . . . . 5 (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
36 eqid 2651 . . . . 5 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
37 uncf2.r . . . . . . 7 (𝜑𝑅 ∈ (𝑋𝐻𝑍))
38 uncf2.s . . . . . . 7 (𝜑𝑆 ∈ (𝑌𝐽𝑊))
39 opelxpi 5182 . . . . . . 7 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → ⟨𝑅, 𝑆⟩ ∈ ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
4037, 38, 39syl2anc 694 . . . . . 6 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
41 uncf2.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
42 uncf2.j . . . . . . 7 𝐽 = (Hom ‘𝐷)
4310, 11, 12, 41, 42, 28, 29, 32, 33, 36xpchom2 16873 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) = ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
4440, 43eleqtrrd 2733 . . . . 5 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))
4513, 24, 27, 31, 35, 36, 44cofu2 16593 . . . 4 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)))
469, 45syl5eq 2697 . . 3 (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))⟨𝑍, 𝑊⟩)𝑆) = ((((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)))
478, 46eqtrd 2685 . 2 (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑍, 𝑊⟩)𝑆) = ((((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)))
4814, 13, 36, 21, 23, 31prf1 16887 . . . . . 6 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩)
4913, 20, 4, 31cofu1 16591 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
5010, 13, 36, 18, 2, 19, 311stf1 16879 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
51 op1stg 7222 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
5228, 29, 51syl2anc 694 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
5350, 52eqtrd 2685 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
5453fveq2d 6233 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
5549, 54eqtrd 2685 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
5610, 13, 36, 18, 2, 22, 312ndf1 16882 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
57 op2ndg 7223 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5828, 29, 57syl2anc 694 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5956, 58eqtrd 2685 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
6055, 59opeq12d 4441 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
6148, 60eqtrd 2685 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
6214, 13, 36, 21, 23, 35prf1 16887 . . . . . 6 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩) = ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩)⟩)
6313, 20, 4, 35cofu1 16591 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)))
6410, 13, 36, 18, 2, 19, 351stf1 16879 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = (1st ‘⟨𝑍, 𝑊⟩))
65 op1stg 7222 . . . . . . . . . . 11 ((𝑍𝐴𝑊𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6632, 33, 65syl2anc 694 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6764, 66eqtrd 2685 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑍)
6867fveq2d 6233 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)) = ((1st𝐺)‘𝑍))
6963, 68eqtrd 2685 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘𝑍))
7010, 13, 36, 18, 2, 22, 352ndf1 16882 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = (2nd ‘⟨𝑍, 𝑊⟩))
71 op2ndg 7223 . . . . . . . . 9 ((𝑍𝐴𝑊𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
7232, 33, 71syl2anc 694 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
7370, 72eqtrd 2685 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑊)
7469, 73opeq12d 4441 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩)⟩ = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7562, 74eqtrd 2685 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩) = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7661, 75oveq12d 6708 . . . 4 (𝜑 → (((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩)) = (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩))
7714, 13, 36, 21, 23, 31, 35, 44prf2 16889 . . . . 5 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ⟨((⟨𝑋, 𝑌⟩(2nd ‘(𝐺func (𝐶 1stF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩), ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)⟩)
7813, 20, 4, 31, 35, 36, 44cofu2 16593 . . . . . . 7 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐺func (𝐶 1stF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd𝐺)((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)))
7953, 67oveq12d 6708 . . . . . . . 8 (𝜑 → (((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd𝐺)((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)) = (𝑋(2nd𝐺)𝑍))
8010, 13, 36, 18, 2, 19, 31, 351stf2 16880 . . . . . . . . . 10 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
8180fveq1d 6231 . . . . . . . . 9 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
82 fvres 6245 . . . . . . . . . 10 (⟨𝑅, 𝑆⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (1st ‘⟨𝑅, 𝑆⟩))
8344, 82syl 17 . . . . . . . . 9 (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (1st ‘⟨𝑅, 𝑆⟩))
84 op1stg 7222 . . . . . . . . . 10 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8537, 38, 84syl2anc 694 . . . . . . . . 9 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8681, 83, 853eqtrd 2689 . . . . . . . 8 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑅)
8779, 86fveq12d 6235 . . . . . . 7 (𝜑 → ((((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd𝐺)((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)) = ((𝑋(2nd𝐺)𝑍)‘𝑅))
8878, 87eqtrd 2685 . . . . . 6 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐺func (𝐶 1stF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((𝑋(2nd𝐺)𝑍)‘𝑅))
8910, 13, 36, 18, 2, 22, 31, 352ndf2 16883 . . . . . . . 8 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩) = (2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
9089fveq1d 6231 . . . . . . 7 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
91 fvres 6245 . . . . . . . 8 (⟨𝑅, 𝑆⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) → ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (2nd ‘⟨𝑅, 𝑆⟩))
9244, 91syl 17 . . . . . . 7 (𝜑 → ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (2nd ‘⟨𝑅, 𝑆⟩))
93 op2ndg 7223 . . . . . . . 8 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
9437, 38, 93syl2anc 694 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
9590, 92, 943eqtrd 2689 . . . . . 6 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑆)
9688, 95opeq12d 4441 . . . . 5 (𝜑 → ⟨((⟨𝑋, 𝑌⟩(2nd ‘(𝐺func (𝐶 1stF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩), ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)⟩ = ⟨((𝑋(2nd𝐺)𝑍)‘𝑅), 𝑆⟩)
9777, 96eqtrd 2685 . . . 4 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ⟨((𝑋(2nd𝐺)𝑍)‘𝑅), 𝑆⟩)
9876, 97fveq12d 6235 . . 3 (𝜑 → ((((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)) = ((⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)‘⟨((𝑋(2nd𝐺)𝑍)‘𝑅), 𝑆⟩))
99 df-ov 6693 . . 3 (((𝑋(2nd𝐺)𝑍)‘𝑅)(⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)𝑆) = ((⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)‘⟨((𝑋(2nd𝐺)𝑍)‘𝑅), 𝑆⟩)
10098, 99syl6eqr 2703 . 2 (𝜑 → ((((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩)(2nd ‘(𝐷 evalF 𝐸))((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩))‘((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩)) = (((𝑋(2nd𝐺)𝑍)‘𝑅)(⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)𝑆))
101 eqid 2651 . . 3 (comp‘𝐸) = (comp‘𝐸)
102 eqid 2651 . . 3 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
10326fucbas 16667 . . . . 5 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
104 relfunc 16569 . . . . . 6 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
105 1st2ndbr 7261 . . . . . 6 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
106104, 4, 105sylancr 696 . . . . 5 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
10711, 103, 106funcf1 16573 . . . 4 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
108107, 28ffvelrnd 6400 . . 3 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
109107, 32ffvelrnd 6400 . . 3 (𝜑 → ((1st𝐺)‘𝑍) ∈ (𝐷 Func 𝐸))
110 eqid 2651 . . 3 (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩) = (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)
11126, 102fuchom 16668 . . . . 5 (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
11211, 41, 111, 106, 28, 32funcf2 16575 . . . 4 (𝜑 → (𝑋(2nd𝐺)𝑍):(𝑋𝐻𝑍)⟶(((1st𝐺)‘𝑋)(𝐷 Nat 𝐸)((1st𝐺)‘𝑍)))
113112, 37ffvelrnd 6400 . . 3 (𝜑 → ((𝑋(2nd𝐺)𝑍)‘𝑅) ∈ (((1st𝐺)‘𝑋)(𝐷 Nat 𝐸)((1st𝐺)‘𝑍)))
11425, 2, 3, 12, 42, 101, 102, 108, 109, 29, 33, 110, 113, 38evlf2val 16906 . 2 (𝜑 → (((𝑋(2nd𝐺)𝑍)‘𝑅)(⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)𝑆) = ((((𝑋(2nd𝐺)𝑍)‘𝑅)‘𝑊)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑌), ((1st ‘((1st𝐺)‘𝑋))‘𝑊)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑍))‘𝑊))((𝑌(2nd ‘((1st𝐺)‘𝑋))𝑊)‘𝑆)))
11547, 100, 1143eqtrd 2689 1 (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑍, 𝑊⟩)𝑆) = ((((𝑋(2nd𝐺)𝑍)‘𝑅)‘𝑊)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑌), ((1st ‘((1st𝐺)‘𝑋))‘𝑊)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑍))‘𝑊))((𝑌(2nd ‘((1st𝐺)‘𝑋))𝑊)‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cop 4216   class class class wbr 4685   × cxp 5141  cres 5145  Rel wrel 5148  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  ⟨“cs3 13633  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372   Func cfunc 16561  func ccofu 16563   Nat cnat 16648   FuncCat cfuc 16649   ×c cxpc 16855   1stF c1stf 16856   2ndF c2ndf 16857   ⟨,⟩F cprf 16858   evalF cevlf 16896   uncurryF cuncf 16898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-func 16565  df-cofu 16567  df-nat 16650  df-fuc 16651  df-xpc 16859  df-1stf 16860  df-2ndf 16861  df-prf 16862  df-evlf 16900  df-uncf 16902
This theorem is referenced by:  curfuncf  16925  uncfcurf  16926
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