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Theorem uncf2 18032
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 18029 . . . . . 6 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6816 . . . . 5 (𝜑 → (2nd𝐹) = (2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 7334 . . . 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 7334 . . 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 7320 . . . 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 2737 . . . . . 6 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11 uncf1.a . . . . . 6 𝐴 = (Base‘𝐶)
12 uncf1.b . . . . . 6 𝐵 = (Base‘𝐷)
1310, 11, 12xpcbas 17972 . . . . 5 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
14 eqid 2737 . . . . . 6 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
15 eqid 2737 . . . . . 6 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
16 funcrcl 17655 . . . . . . . . . 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 495 . . . . . . . 8 (𝜑𝐶 ∈ Cat)
19 eqid 2737 . . . . . . . 8 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
2010, 18, 2, 191stfcl 17991 . . . . . . 7 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2120, 4cofucl 17680 . . . . . 6 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
22 eqid 2737 . . . . . . 7 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
2310, 18, 2, 222ndfcl 17992 . . . . . 6 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2414, 15, 21, 23prfcl 17997 . . . . 5 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
25 eqid 2737 . . . . . 6 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
26 eqid 2737 . . . . . 6 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2725, 26, 2, 3evlfcl 18017 . . . . 5 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
28 uncf1.x . . . . . 6 (𝜑𝑋𝐴)
29 uncf1.y . . . . . 6 (𝜑𝑌𝐵)
3028, 29opelxpd 5646 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
31 uncf2.z . . . . . 6 (𝜑𝑍𝐴)
32 uncf2.w . . . . . 6 (𝜑𝑊𝐵)
3331, 32opelxpd 5646 . . . . 5 (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
34 eqid 2737 . . . . 5 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
35 uncf2.r . . . . . . 7 (𝜑𝑅 ∈ (𝑋𝐻𝑍))
36 uncf2.s . . . . . . 7 (𝜑𝑆 ∈ (𝑌𝐽𝑊))
3735, 36opelxpd 5646 . . . . . 6 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
38 uncf2.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
39 uncf2.j . . . . . . 7 𝐽 = (Hom ‘𝐷)
4010, 11, 12, 38, 39, 28, 29, 31, 32, 34xpchom2 17980 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) = ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
4137, 40eleqtrrd 2841 . . . . 5 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))
4213, 24, 27, 30, 33, 34, 41cofu2 17678 . . . 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⟩)))
439, 42eqtrid 2789 . . 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⟩)))
448, 43eqtrd 2777 . 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⟩)))
4514, 13, 34, 21, 23, 30prf1 17994 . . . . . 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⟩)⟩)
4613, 20, 4, 30cofu1 17676 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
4710, 13, 34, 18, 2, 19, 301stf1 17986 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
48 op1stg 7890 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
4928, 29, 48syl2anc 584 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
5047, 49eqtrd 2777 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
5150fveq2d 6816 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
5246, 51eqtrd 2777 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
5310, 13, 34, 18, 2, 22, 302ndf1 17989 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
54 op2ndg 7891 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5528, 29, 54syl2anc 584 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5653, 55eqtrd 2777 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
5752, 56opeq12d 4823 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
5845, 57eqtrd 2777 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
5914, 13, 34, 21, 23, 33prf1 17994 . . . . . 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⟩)⟩)
6013, 20, 4, 33cofu1 17676 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)))
6110, 13, 34, 18, 2, 19, 331stf1 17986 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = (1st ‘⟨𝑍, 𝑊⟩))
62 op1stg 7890 . . . . . . . . . . 11 ((𝑍𝐴𝑊𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6331, 32, 62syl2anc 584 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6461, 63eqtrd 2777 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑍)
6564fveq2d 6816 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)) = ((1st𝐺)‘𝑍))
6660, 65eqtrd 2777 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘𝑍))
6710, 13, 34, 18, 2, 22, 332ndf1 17989 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = (2nd ‘⟨𝑍, 𝑊⟩))
68 op2ndg 7891 . . . . . . . . 9 ((𝑍𝐴𝑊𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
6931, 32, 68syl2anc 584 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
7067, 69eqtrd 2777 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑊)
7166, 70opeq12d 4823 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩)⟩ = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7259, 71eqtrd 2777 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩) = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7358, 72oveq12d 7335 . . . 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⟩))
7414, 13, 34, 21, 23, 30, 33, 41prf2 17996 . . . . 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⟩)⟩)
7513, 20, 4, 30, 33, 34, 41cofu2 17678 . . . . . . 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⟩)))
7650, 64oveq12d 7335 . . . . . . . 8 (𝜑 → (((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)(2nd𝐺)((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)) = (𝑋(2nd𝐺)𝑍))
7710, 13, 34, 18, 2, 19, 30, 331stf2 17987 . . . . . . . . . 10 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
7877fveq1d 6814 . . . . . . . . 9 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
7941fvresd 6832 . . . . . . . . 9 (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (1st ‘⟨𝑅, 𝑆⟩))
80 op1stg 7890 . . . . . . . . . 10 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8135, 36, 80syl2anc 584 . . . . . . . . 9 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8278, 79, 813eqtrd 2781 . . . . . . . 8 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑅)
8376, 82fveq12d 6819 . . . . . . 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))
8475, 83eqtrd 2777 . . . . . 6 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐺func (𝐶 1stF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((𝑋(2nd𝐺)𝑍)‘𝑅))
8510, 13, 34, 18, 2, 22, 30, 332ndf2 17990 . . . . . . . 8 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩) = (2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
8685fveq1d 6814 . . . . . . 7 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
8741fvresd 6832 . . . . . . 7 (𝜑 → ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (2nd ‘⟨𝑅, 𝑆⟩))
88 op2ndg 7891 . . . . . . . 8 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
8935, 36, 88syl2anc 584 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
9086, 87, 893eqtrd 2781 . . . . . 6 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑆)
9184, 90opeq12d 4823 . . . . 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⟩)
9274, 91eqtrd 2777 . . . 4 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ⟨((𝑋(2nd𝐺)𝑍)‘𝑅), 𝑆⟩)
9373, 92fveq12d 6819 . . 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⟩))
94 df-ov 7320 . . 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⟩)
9593, 94eqtr4di 2795 . 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))
96 eqid 2737 . . 3 (comp‘𝐸) = (comp‘𝐸)
97 eqid 2737 . . 3 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
9826fucbas 17754 . . . . 5 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
99 relfunc 17654 . . . . . 6 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
100 1st2ndbr 7930 . . . . . 6 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
10199, 4, 100sylancr 587 . . . . 5 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
10211, 98, 101funcf1 17658 . . . 4 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
103102, 28ffvelcdmd 7002 . . 3 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
104102, 31ffvelcdmd 7002 . . 3 (𝜑 → ((1st𝐺)‘𝑍) ∈ (𝐷 Func 𝐸))
105 eqid 2737 . . 3 (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩) = (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)
10626, 97fuchom 17755 . . . . 5 (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
10711, 38, 106, 101, 28, 31funcf2 17660 . . . 4 (𝜑 → (𝑋(2nd𝐺)𝑍):(𝑋𝐻𝑍)⟶(((1st𝐺)‘𝑋)(𝐷 Nat 𝐸)((1st𝐺)‘𝑍)))
108107, 35ffvelcdmd 7002 . . 3 (𝜑 → ((𝑋(2nd𝐺)𝑍)‘𝑅) ∈ (((1st𝐺)‘𝑋)(𝐷 Nat 𝐸)((1st𝐺)‘𝑍)))
10925, 2, 3, 12, 39, 96, 97, 103, 104, 29, 32, 105, 108, 36evlf2val 18014 . 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)))
11044, 95, 1093eqtrd 2781 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 396   = wceq 1540  wcel 2105  cop 4577   class class class wbr 5087   × cxp 5606  cres 5610  Rel wrel 5613  cfv 6466  (class class class)co 7317  1st c1st 7876  2nd c2nd 7877  ⟨“cs3 14634  Basecbs 16989  Hom chom 17050  compcco 17051  Catccat 17450   Func cfunc 17646  func ccofu 17648   Nat cnat 17734   FuncCat cfuc 17735   ×c cxpc 17962   1stF c1stf 17963   2ndF c2ndf 17964   ⟨,⟩F cprf 17965   evalF cevlf 18004   uncurryF cuncf 18006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630  ax-cnex 11007  ax-resscn 11008  ax-1cn 11009  ax-icn 11010  ax-addcl 11011  ax-addrcl 11012  ax-mulcl 11013  ax-mulrcl 11014  ax-mulcom 11015  ax-addass 11016  ax-mulass 11017  ax-distr 11018  ax-i2m1 11019  ax-1ne0 11020  ax-1rid 11021  ax-rnegex 11022  ax-rrecex 11023  ax-cnre 11024  ax-pre-lttri 11025  ax-pre-lttrn 11026  ax-pre-ltadd 11027  ax-pre-mulgt0 11028
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-pred 6225  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-riota 7274  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-1st 7878  df-2nd 7879  df-frecs 8146  df-wrecs 8177  df-recs 8251  df-rdg 8290  df-1o 8346  df-er 8548  df-map 8667  df-ixp 8736  df-en 8784  df-dom 8785  df-sdom 8786  df-fin 8787  df-card 9775  df-pnf 11091  df-mnf 11092  df-xr 11093  df-ltxr 11094  df-le 11095  df-sub 11287  df-neg 11288  df-nn 12054  df-2 12116  df-3 12117  df-4 12118  df-5 12119  df-6 12120  df-7 12121  df-8 12122  df-9 12123  df-n0 12314  df-z 12400  df-dec 12518  df-uz 12663  df-fz 13320  df-fzo 13463  df-hash 14125  df-word 14297  df-concat 14353  df-s1 14380  df-s2 14640  df-s3 14641  df-struct 16925  df-slot 16960  df-ndx 16972  df-base 16990  df-hom 17063  df-cco 17064  df-cat 17454  df-cid 17455  df-func 17650  df-cofu 17652  df-nat 17736  df-fuc 17737  df-xpc 17966  df-1stf 17967  df-2ndf 17968  df-prf 17969  df-evlf 18008  df-uncf 18010
This theorem is referenced by:  curfuncf  18033  uncfcurf  18034
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