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Theorem uncf2 17085
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 17082 . . . . . 6 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6336 . . . . 5 (𝜑 → (2nd𝐹) = (2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 6810 . . . 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 6810 . . 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 6796 . . . 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 2771 . . . . . 6 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11 uncf1.a . . . . . 6 𝐴 = (Base‘𝐶)
12 uncf1.b . . . . . 6 𝐵 = (Base‘𝐷)
1310, 11, 12xpcbas 17026 . . . . 5 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
14 eqid 2771 . . . . . 6 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
15 eqid 2771 . . . . . 6 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
16 funcrcl 16730 . . . . . . . . . 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 482 . . . . . . . 8 (𝜑𝐶 ∈ Cat)
19 eqid 2771 . . . . . . . 8 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
2010, 18, 2, 191stfcl 17045 . . . . . . 7 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2120, 4cofucl 16755 . . . . . 6 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
22 eqid 2771 . . . . . . 7 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
2310, 18, 2, 222ndfcl 17046 . . . . . 6 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2414, 15, 21, 23prfcl 17051 . . . . 5 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
25 eqid 2771 . . . . . 6 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
26 eqid 2771 . . . . . 6 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2725, 26, 2, 3evlfcl 17070 . . . . 5 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
28 uncf1.x . . . . . 6 (𝜑𝑋𝐴)
29 uncf1.y . . . . . 6 (𝜑𝑌𝐵)
30 opelxpi 5288 . . . . . 6 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
3128, 29, 30syl2anc 573 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
32 uncf2.z . . . . . 6 (𝜑𝑍𝐴)
33 uncf2.w . . . . . 6 (𝜑𝑊𝐵)
34 opelxpi 5288 . . . . . 6 ((𝑍𝐴𝑊𝐵) → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
3532, 33, 34syl2anc 573 . . . . 5 (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
36 eqid 2771 . . . . 5 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
37 uncf2.r . . . . . . 7 (𝜑𝑅 ∈ (𝑋𝐻𝑍))
38 uncf2.s . . . . . . 7 (𝜑𝑆 ∈ (𝑌𝐽𝑊))
39 opelxpi 5288 . . . . . . 7 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → ⟨𝑅, 𝑆⟩ ∈ ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
4037, 38, 39syl2anc 573 . . . . . 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 17034 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) = ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
4440, 43eleqtrrd 2853 . . . . 5 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))
4513, 24, 27, 31, 35, 36, 44cofu2 16753 . . . 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 2817 . . 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 2805 . 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 17048 . . . . . 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 16751 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
5010, 13, 36, 18, 2, 19, 311stf1 17040 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
51 op1stg 7327 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
5228, 29, 51syl2anc 573 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
5350, 52eqtrd 2805 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
5453fveq2d 6336 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
5549, 54eqtrd 2805 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
5610, 13, 36, 18, 2, 22, 312ndf1 17043 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
57 op2ndg 7328 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5828, 29, 57syl2anc 573 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5956, 58eqtrd 2805 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
6055, 59opeq12d 4547 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
6148, 60eqtrd 2805 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
6214, 13, 36, 21, 23, 35prf1 17048 . . . . . 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 16751 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)))
6410, 13, 36, 18, 2, 19, 351stf1 17040 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = (1st ‘⟨𝑍, 𝑊⟩))
65 op1stg 7327 . . . . . . . . . . 11 ((𝑍𝐴𝑊𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6632, 33, 65syl2anc 573 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6764, 66eqtrd 2805 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑍)
6867fveq2d 6336 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)) = ((1st𝐺)‘𝑍))
6963, 68eqtrd 2805 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘𝑍))
7010, 13, 36, 18, 2, 22, 352ndf1 17043 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = (2nd ‘⟨𝑍, 𝑊⟩))
71 op2ndg 7328 . . . . . . . . 9 ((𝑍𝐴𝑊𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
7232, 33, 71syl2anc 573 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
7370, 72eqtrd 2805 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑊)
7469, 73opeq12d 4547 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩)⟩ = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7562, 74eqtrd 2805 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩) = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7661, 75oveq12d 6811 . . . 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 17050 . . . . 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 16753 . . . . . . 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 6811 . . . . . . . 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 17041 . . . . . . . . . 10 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
8180fveq1d 6334 . . . . . . . . 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 6348 . . . . . . . . . 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 7327 . . . . . . . . . 10 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8537, 38, 84syl2anc 573 . . . . . . . . 9 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8681, 83, 853eqtrd 2809 . . . . . . . 8 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑅)
8779, 86fveq12d 6338 . . . . . . 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 2805 . . . . . 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 17044 . . . . . . . 8 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩) = (2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
9089fveq1d 6334 . . . . . . 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 6348 . . . . . . . 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 7328 . . . . . . . 8 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
9437, 38, 93syl2anc 573 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
9590, 92, 943eqtrd 2809 . . . . . 6 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑆)
9688, 95opeq12d 4547 . . . . 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 2805 . . . 4 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ⟨((𝑋(2nd𝐺)𝑍)‘𝑅), 𝑆⟩)
9876, 97fveq12d 6338 . . 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 6796 . . 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 2823 . 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 2771 . . 3 (comp‘𝐸) = (comp‘𝐸)
102 eqid 2771 . . 3 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
10326fucbas 16827 . . . . 5 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
104 relfunc 16729 . . . . . 6 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
105 1st2ndbr 7366 . . . . . 6 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
106104, 4, 105sylancr 575 . . . . 5 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
10711, 103, 106funcf1 16733 . . . 4 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
108107, 28ffvelrnd 6503 . . 3 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
109107, 32ffvelrnd 6503 . . 3 (𝜑 → ((1st𝐺)‘𝑍) ∈ (𝐷 Func 𝐸))
110 eqid 2771 . . 3 (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩) = (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)
11126, 102fuchom 16828 . . . . 5 (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
11211, 41, 111, 106, 28, 32funcf2 16735 . . . 4 (𝜑 → (𝑋(2nd𝐺)𝑍):(𝑋𝐻𝑍)⟶(((1st𝐺)‘𝑋)(𝐷 Nat 𝐸)((1st𝐺)‘𝑍)))
113112, 37ffvelrnd 6503 . . 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 17067 . 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 2809 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 382   = wceq 1631  wcel 2145  cop 4322   class class class wbr 4786   × cxp 5247  cres 5251  Rel wrel 5254  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  ⟨“cs3 13796  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532   Func cfunc 16721  func ccofu 16723   Nat cnat 16808   FuncCat cfuc 16809   ×c cxpc 17016   1stF c1stf 17017   2ndF c2ndf 17018   ⟨,⟩F cprf 17019   evalF cevlf 17057   uncurryF cuncf 17059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-s2 13802  df-s3 13803  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-hom 16174  df-cco 16175  df-cat 16536  df-cid 16537  df-func 16725  df-cofu 16727  df-nat 16810  df-fuc 16811  df-xpc 17020  df-1stf 17021  df-2ndf 17022  df-prf 17023  df-evlf 17061  df-uncf 17063
This theorem is referenced by:  curfuncf  17086  uncfcurf  17087
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