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Theorem uncf2 18126
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 18123 . . . . . 6 (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
65fveq2d 6846 . . . . 5 (𝜑 → (2nd𝐹) = (2nd ‘((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))))
76oveqd 7374 . . . 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 7374 . . 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 7360 . . . 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 2736 . . . . . 6 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11 uncf1.a . . . . . 6 𝐴 = (Base‘𝐶)
12 uncf1.b . . . . . 6 𝐵 = (Base‘𝐷)
1310, 11, 12xpcbas 18066 . . . . 5 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
14 eqid 2736 . . . . . 6 ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))
15 eqid 2736 . . . . . 6 ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷)
16 funcrcl 17749 . . . . . . . . . 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 2736 . . . . . . . 8 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
2010, 18, 2, 191stfcl 18085 . . . . . . 7 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
2120, 4cofucl 17774 . . . . . 6 (𝜑 → (𝐺func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸)))
22 eqid 2736 . . . . . . 7 (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷)
2310, 18, 2, 222ndfcl 18086 . . . . . 6 (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷))
2414, 15, 21, 23prfcl 18091 . . . . 5 (𝜑 → ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷)))
25 eqid 2736 . . . . . 6 (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸)
26 eqid 2736 . . . . . 6 (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2725, 26, 2, 3evlfcl 18111 . . . . 5 (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸))
28 uncf1.x . . . . . 6 (𝜑𝑋𝐴)
29 uncf1.y . . . . . 6 (𝜑𝑌𝐵)
3028, 29opelxpd 5671 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
31 uncf2.z . . . . . 6 (𝜑𝑍𝐴)
32 uncf2.w . . . . . 6 (𝜑𝑊𝐵)
3331, 32opelxpd 5671 . . . . 5 (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐴 × 𝐵))
34 eqid 2736 . . . . 5 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
35 uncf2.r . . . . . . 7 (𝜑𝑅 ∈ (𝑋𝐻𝑍))
36 uncf2.s . . . . . . 7 (𝜑𝑆 ∈ (𝑌𝐽𝑊))
3735, 36opelxpd 5671 . . . . . 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 18074 . . . . . 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 17772 . . . 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 2788 . . 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 2776 . 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 18088 . . . . . 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 17770 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)))
4710, 13, 34, 18, 2, 19, 301stf1 18080 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
48 op1stg 7933 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
4928, 29, 48syl2anc 584 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
5047, 49eqtrd 2776 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑋)
5150fveq2d 6846 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st𝐺)‘𝑋))
5246, 51eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st𝐺)‘𝑋))
5310, 13, 34, 18, 2, 22, 302ndf1 18083 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = (2nd ‘⟨𝑋, 𝑌⟩))
54 op2ndg 7934 . . . . . . . . 9 ((𝑋𝐴𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5528, 29, 54syl2anc 584 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
5653, 55eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩) = 𝑌)
5752, 56opeq12d 4838 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑋, 𝑌⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑋, 𝑌⟩)⟩ = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
5845, 57eqtrd 2776 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑋, 𝑌⟩) = ⟨((1st𝐺)‘𝑋), 𝑌⟩)
5914, 13, 34, 21, 23, 33prf1 18088 . . . . . 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 17770 . . . . . . . 8 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)))
6110, 13, 34, 18, 2, 19, 331stf1 18080 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = (1st ‘⟨𝑍, 𝑊⟩))
62 op1stg 7933 . . . . . . . . . . 11 ((𝑍𝐴𝑊𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6331, 32, 62syl2anc 584 . . . . . . . . . 10 (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
6461, 63eqtrd 2776 . . . . . . . . 9 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑍)
6564fveq2d 6846 . . . . . . . 8 (𝜑 → ((1st𝐺)‘((1st ‘(𝐶 1stF 𝐷))‘⟨𝑍, 𝑊⟩)) = ((1st𝐺)‘𝑍))
6660, 65eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩) = ((1st𝐺)‘𝑍))
6710, 13, 34, 18, 2, 22, 332ndf1 18083 . . . . . . . 8 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = (2nd ‘⟨𝑍, 𝑊⟩))
68 op2ndg 7934 . . . . . . . . 9 ((𝑍𝐴𝑊𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
6931, 32, 68syl2anc 584 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
7067, 69eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩) = 𝑊)
7166, 70opeq12d 4838 . . . . . 6 (𝜑 → ⟨((1st ‘(𝐺func (𝐶 1stF 𝐷)))‘⟨𝑍, 𝑊⟩), ((1st ‘(𝐶 2ndF 𝐷))‘⟨𝑍, 𝑊⟩)⟩ = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7259, 71eqtrd 2776 . . . . 5 (𝜑 → ((1st ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))‘⟨𝑍, 𝑊⟩) = ⟨((1st𝐺)‘𝑍), 𝑊⟩)
7358, 72oveq12d 7375 . . . 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 18090 . . . . 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 17772 . . . . . . 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 7375 . . . . . . . 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 18081 . . . . . . . . . 10 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩) = (1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
7877fveq1d 6844 . . . . . . . . 9 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
7941fvresd 6862 . . . . . . . . 9 (𝜑 → ((1st ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (1st ‘⟨𝑅, 𝑆⟩))
80 op1stg 7933 . . . . . . . . . 10 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8135, 36, 80syl2anc 584 . . . . . . . . 9 (𝜑 → (1st ‘⟨𝑅, 𝑆⟩) = 𝑅)
8278, 79, 813eqtrd 2780 . . . . . . . 8 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑅)
8376, 82fveq12d 6849 . . . . . . 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 2776 . . . . . 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 18084 . . . . . . . 8 (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩) = (2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩)))
8685fveq1d 6844 . . . . . . 7 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩))
8741fvresd 6862 . . . . . . 7 (𝜑 → ((2nd ↾ (⟨𝑋, 𝑌⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑍, 𝑊⟩))‘⟨𝑅, 𝑆⟩) = (2nd ‘⟨𝑅, 𝑆⟩))
88 op2ndg 7934 . . . . . . . 8 ((𝑅 ∈ (𝑋𝐻𝑍) ∧ 𝑆 ∈ (𝑌𝐽𝑊)) → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
8935, 36, 88syl2anc 584 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑅, 𝑆⟩) = 𝑆)
9086, 87, 893eqtrd 2780 . . . . . 6 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘(𝐶 2ndF 𝐷))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = 𝑆)
9184, 90opeq12d 4838 . . . . 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 2776 . . . 4 (𝜑 → ((⟨𝑋, 𝑌⟩(2nd ‘((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))⟨𝑍, 𝑊⟩)‘⟨𝑅, 𝑆⟩) = ⟨((𝑋(2nd𝐺)𝑍)‘𝑅), 𝑆⟩)
9373, 92fveq12d 6849 . . 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 7360 . . 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 2794 . 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 2736 . . 3 (comp‘𝐸) = (comp‘𝐸)
97 eqid 2736 . . 3 (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
9826fucbas 17848 . . . . 5 (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
99 relfunc 17748 . . . . . 6 Rel (𝐶 Func (𝐷 FuncCat 𝐸))
100 1st2ndbr 7974 . . . . . 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 17752 . . . 4 (𝜑 → (1st𝐺):𝐴⟶(𝐷 Func 𝐸))
103102, 28ffvelcdmd 7036 . . 3 (𝜑 → ((1st𝐺)‘𝑋) ∈ (𝐷 Func 𝐸))
104102, 31ffvelcdmd 7036 . . 3 (𝜑 → ((1st𝐺)‘𝑍) ∈ (𝐷 Func 𝐸))
105 eqid 2736 . . 3 (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩) = (⟨((1st𝐺)‘𝑋), 𝑌⟩(2nd ‘(𝐷 evalF 𝐸))⟨((1st𝐺)‘𝑍), 𝑊⟩)
10626, 97fuchom 17849 . . . . 5 (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
10711, 38, 106, 101, 28, 31funcf2 17754 . . . 4 (𝜑 → (𝑋(2nd𝐺)𝑍):(𝑋𝐻𝑍)⟶(((1st𝐺)‘𝑋)(𝐷 Nat 𝐸)((1st𝐺)‘𝑍)))
108107, 35ffvelcdmd 7036 . . 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 18108 . 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 2780 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 1541  wcel 2106  cop 4592   class class class wbr 5105   × cxp 5631  cres 5635  Rel wrel 5638  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  ⟨“cs3 14731  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544   Func cfunc 17740  func ccofu 17742   Nat cnat 17828   FuncCat cfuc 17829   ×c cxpc 18056   1stF c1stf 18057   2ndF c2ndf 18058   ⟨,⟩F cprf 18059   evalF cevlf 18098   uncurryF cuncf 18100
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-s2 14737  df-s3 14738  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-func 17744  df-cofu 17746  df-nat 17830  df-fuc 17831  df-xpc 18060  df-1stf 18061  df-2ndf 18062  df-prf 18063  df-evlf 18102  df-uncf 18104
This theorem is referenced by:  curfuncf  18127  uncfcurf  18128
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