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Theorem uncf2 17479
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 17476 . . . . . 6 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6649 . . . . 5 (𝜑 → (2nd𝐹) = (2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 7152 . . . 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 7152 . . 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 7138 . . . 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 2798 . . . . . 6 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11 uncf1.a . . . . . 6 𝐴 = (Base‘𝐶)
12 uncf1.b . . . . . 6 𝐵 = (Base‘𝐷)
1310, 11, 12xpcbas 17420 . . . . 5 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
14 eqid 2798 . . . . . 6 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
15 eqid 2798 . . . . . 6 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
16 funcrcl 17125 . . . . . . . . . 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 498 . . . . . . . 8 (𝜑𝐶 ∈ Cat)
19 eqid 2798 . . . . . . . 8 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
2010, 18, 2, 191stfcl 17439 . . . . . . 7 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2120, 4cofucl 17150 . . . . . 6 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
22 eqid 2798 . . . . . . 7 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
2310, 18, 2, 222ndfcl 17440 . . . . . 6 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2414, 15, 21, 23prfcl 17445 . . . . 5 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
25 eqid 2798 . . . . . 6 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
26 eqid 2798 . . . . . 6 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2725, 26, 2, 3evlfcl 17464 . . . . 5 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
28 uncf1.x . . . . . 6 (𝜑𝑋𝐴)
29 uncf1.y . . . . . 6 (𝜑𝑌𝐵)
3028, 29opelxpd 5557 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
31 uncf2.z . . . . . 6 (𝜑𝑍𝐴)
32 uncf2.w . . . . . 6 (𝜑𝑊𝐵)
3331, 32opelxpd 5557 . . . . 5 (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
34 eqid 2798 . . . . 5 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
35 uncf2.r . . . . . . 7 (𝜑𝑅 ∈ (𝑋𝐻𝑍))
36 uncf2.s . . . . . . 7 (𝜑𝑆 ∈ (𝑌𝐽𝑊))
3735, 36opelxpd 5557 . . . . . 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 17428 . . . . . 6 (𝜑 → (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩) = ((𝑋𝐻𝑍) × (𝑌𝐽𝑊)))
4137, 40eleqtrrd 2893 . . . . 5 (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))
4213, 24, 27, 30, 33, 34, 41cofu2 17148 . . . 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, 42syl5eq 2845 . . 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 2833 . 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 17442 . . . . . 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 17146 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
4710, 13, 34, 18, 2, 19, 301stf1 17434 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
48 op1stg 7683 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
4928, 29, 48syl2anc 587 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
5047, 49eqtrd 2833 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
5150fveq2d 6649 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
5246, 51eqtrd 2833 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
5310, 13, 34, 18, 2, 22, 302ndf1 17437 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
54 op2ndg 7684 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5528, 29, 54syl2anc 587 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5653, 55eqtrd 2833 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
5752, 56opeq12d 4773 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
5845, 57eqtrd 2833 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
5914, 13, 34, 21, 23, 33prf1 17442 . . . . . 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 17146 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)))
6110, 13, 34, 18, 2, 19, 331stf1 17434 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = (1st ‘⟨𝑍, 𝑊⟩))
62 op1stg 7683 . . . . . . . . . . 11 ((𝑍𝐴𝑊𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6331, 32, 62syl2anc 587 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6461, 63eqtrd 2833 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑍)
6564fveq2d 6649 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)) = ((1st𝐺)‘𝑍))
6660, 65eqtrd 2833 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘𝑍))
6710, 13, 34, 18, 2, 22, 332ndf1 17437 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = (2nd ‘⟨𝑍, 𝑊⟩))
68 op2ndg 7684 . . . . . . . . 9 ((𝑍𝐴𝑊𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
6931, 32, 68syl2anc 587 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
7067, 69eqtrd 2833 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑊)
7166, 70opeq12d 4773 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩)⟩ = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7259, 71eqtrd 2833 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩) = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7358, 72oveq12d 7153 . . . 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 17444 . . . . 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 17148 . . . . . . 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 7153 . . . . . . . 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 17435 . . . . . . . . . 10 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
7877fveq1d 6647 . . . . . . . . 9 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
7941fvresd 6665 . . . . . . . . 9 (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (1st ‘⟨𝑅, 𝑆⟩))
80 op1stg 7683 . . . . . . . . . 10 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8135, 36, 80syl2anc 587 . . . . . . . . 9 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8278, 79, 813eqtrd 2837 . . . . . . . 8 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑅)
8376, 82fveq12d 6652 . . . . . . 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 2833 . . . . . 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 17438 . . . . . . . 8 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩) = (2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
8685fveq1d 6647 . . . . . . 7 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
8741fvresd 6665 . . . . . . 7 (𝜑 → ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (2nd ‘⟨𝑅, 𝑆⟩))
88 op2ndg 7684 . . . . . . . 8 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
8935, 36, 88syl2anc 587 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
9086, 87, 893eqtrd 2837 . . . . . 6 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑆)
9184, 90opeq12d 4773 . . . . 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 2833 . . . 4 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ⟨((𝑋(2nd𝐺)𝑍)‘𝑅), 𝑆⟩)
9373, 92fveq12d 6652 . . 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 7138 . . 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 2851 . 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 2798 . . 3 (comp‘𝐸) = (comp‘𝐸)
97 eqid 2798 . . 3 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
9826fucbas 17222 . . . . 5 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
99 relfunc 17124 . . . . . 6 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
100 1st2ndbr 7723 . . . . . 6 ((Rel (𝐶 Func (𝐷 FuncCat 𝐸)) ∧ 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
10199, 4, 100sylancr 590 . . . . 5 (𝜑 → (1st𝐺)(𝐶 Func (𝐷 FuncCat 𝐸))(2nd𝐺))
10211, 98, 101funcf1 17128 . . . 4 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
103102, 28ffvelrnd 6829 . . 3 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
104102, 31ffvelrnd 6829 . . 3 (𝜑 → ((1st𝐺)‘𝑍) ∈ (𝐷 Func 𝐸))
105 eqid 2798 . . 3 (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩) = (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)
10626, 97fuchom 17223 . . . . 5 (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
10711, 38, 106, 101, 28, 31funcf2 17130 . . . 4 (𝜑 → (𝑋(2nd𝐺)𝑍):(𝑋𝐻𝑍)⟶(((1st𝐺)‘𝑋)(𝐷 Nat 𝐸)((1st𝐺)‘𝑍)))
108107, 35ffvelrnd 6829 . . 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 17461 . 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 2837 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 399   = wceq 1538  wcel 2111  cop 4531   class class class wbr 5030   × cxp 5517  cres 5521  Rel wrel 5524  cfv 6324  (class class class)co 7135  1st c1st 7669  2nd c2nd 7670  ⟨“cs3 14195  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927   Func cfunc 17116  func ccofu 17118   Nat cnat 17203   FuncCat cfuc 17204   ×c cxpc 17410   1stF c1stf 17411   2ndF c2ndf 17412   ⟨,⟩F cprf 17413   evalF cevlf 17451   uncurryF cuncf 17453
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-s2 14201  df-s3 14202  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-hom 16581  df-cco 16582  df-cat 16931  df-cid 16932  df-func 17120  df-cofu 17122  df-nat 17205  df-fuc 17206  df-xpc 17414  df-1stf 17415  df-2ndf 17416  df-prf 17417  df-evlf 17455  df-uncf 17457
This theorem is referenced by:  curfuncf  17480  uncfcurf  17481
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