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Theorem yonedalem22 18334
Description: Lemma for yoneda 18339. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y 𝑌 = (Yon‘𝐶)
yoneda.b 𝐵 = (Base‘𝐶)
yoneda.1 1 = (Id‘𝐶)
yoneda.o 𝑂 = (oppCat‘𝐶)
yoneda.s 𝑆 = (SetCat‘𝑈)
yoneda.t 𝑇 = (SetCat‘𝑉)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomF𝑄)
yoneda.r 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (𝜑𝐶 ∈ Cat)
yoneda.w (𝜑𝑉𝑊)
yoneda.u (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoneda.v (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
yonedalem21.f (𝜑𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x (𝜑𝑋𝐵)
yonedalem22.g (𝜑𝐺 ∈ (𝑂 Func 𝑆))
yonedalem22.p (𝜑𝑃𝐵)
yonedalem22.a (𝜑𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
yonedalem22.k (𝜑𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))
Assertion
Ref Expression
yonedalem22 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))

Proof of Theorem yonedalem22
StepHypRef Expression
1 yoneda.z . . . . . . 7 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
21fveq2i 6909 . . . . . 6 (2nd𝑍) = (2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
32oveqi 7443 . . . . 5 (⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩) = (⟨𝐹, 𝑋⟩(2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)
43oveqi 7443 . . . 4 (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (𝐴(⟨𝐹, 𝑋⟩(2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)𝐾)
5 df-ov 7433 . . . 4 (𝐴(⟨𝐹, 𝑋⟩(2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)
64, 5eqtri 2762 . . 3 (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)
7 eqid 2734 . . . . 5 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
8 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
98fucbas 18015 . . . . 5 (𝑂 Func 𝑆) = (Base‘𝑄)
10 yoneda.o . . . . . 6 𝑂 = (oppCat‘𝐶)
11 yoneda.b . . . . . 6 𝐵 = (Base‘𝐶)
1210, 11oppcbas 17763 . . . . 5 𝐵 = (Base‘𝑂)
137, 9, 12xpcbas 18233 . . . 4 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
14 eqid 2734 . . . . 5 ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) = ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))
15 eqid 2734 . . . . 5 ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
16 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
1710oppccat 17768 . . . . . . . . 9 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
1816, 17syl 17 . . . . . . . 8 (𝜑𝑂 ∈ Cat)
19 yoneda.w . . . . . . . . . 10 (𝜑𝑉𝑊)
20 yoneda.v . . . . . . . . . . 11 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2120unssbd 4203 . . . . . . . . . 10 (𝜑𝑈𝑉)
2219, 21ssexd 5329 . . . . . . . . 9 (𝜑𝑈 ∈ V)
23 yoneda.s . . . . . . . . . 10 𝑆 = (SetCat‘𝑈)
2423setccat 18138 . . . . . . . . 9 (𝑈 ∈ V → 𝑆 ∈ Cat)
2522, 24syl 17 . . . . . . . 8 (𝜑𝑆 ∈ Cat)
268, 18, 25fuccat 18026 . . . . . . 7 (𝜑𝑄 ∈ Cat)
27 eqid 2734 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
287, 26, 18, 272ndfcl 18253 . . . . . 6 (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
29 eqid 2734 . . . . . . . 8 (oppCat‘𝑄) = (oppCat‘𝑄)
30 relfunc 17912 . . . . . . . . 9 Rel (𝐶 Func 𝑄)
31 yoneda.y . . . . . . . . . 10 𝑌 = (Yon‘𝐶)
32 yoneda.u . . . . . . . . . 10 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
3331, 16, 10, 23, 8, 22, 32yoncl 18318 . . . . . . . . 9 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
34 1st2ndbr 8065 . . . . . . . . 9 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3530, 33, 34sylancr 587 . . . . . . . 8 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3610, 29, 35funcoppc 17925 . . . . . . 7 (𝜑 → (1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌))
37 df-br 5148 . . . . . . 7 ((1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌) ↔ ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3836, 37sylib 218 . . . . . 6 (𝜑 → ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3928, 38cofucl 17938 . . . . 5 (𝜑 → (⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
40 eqid 2734 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
417, 26, 18, 401stfcl 18252 . . . . 5 (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
4214, 15, 39, 41prfcl 18258 . . . 4 (𝜑 → ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func ((oppCat‘𝑄) ×c 𝑄)))
43 yoneda.h . . . . 5 𝐻 = (HomF𝑄)
44 yoneda.t . . . . 5 𝑇 = (SetCat‘𝑉)
4520unssad 4202 . . . . 5 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
4643, 29, 44, 26, 19, 45hofcl 18315 . . . 4 (𝜑𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
47 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
48 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
4947, 48opelxpd 5727 . . . 4 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
50 yonedalem22.g . . . . 5 (𝜑𝐺 ∈ (𝑂 Func 𝑆))
51 yonedalem22.p . . . . 5 (𝜑𝑃𝐵)
5250, 51opelxpd 5727 . . . 4 (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
53 eqid 2734 . . . 4 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
54 yonedalem22.a . . . . . 6 (𝜑𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
55 yonedalem22.k . . . . . . 7 (𝜑𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))
56 eqid 2734 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
5756, 10oppchom 17760 . . . . . . 7 (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
5855, 57eleqtrrdi 2849 . . . . . 6 (𝜑𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
5954, 58opelxpd 5727 . . . . 5 (𝜑 → ⟨𝐴, 𝐾⟩ ∈ ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
60 eqid 2734 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
618, 60fuchom 18016 . . . . . 6 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
62 eqid 2734 . . . . . 6 (Hom ‘𝑂) = (Hom ‘𝑂)
637, 9, 12, 61, 62, 47, 48, 50, 51, 53xpchom2 18241 . . . . 5 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
6459, 63eleqtrrd 2841 . . . 4 (𝜑 → ⟨𝐴, 𝐾⟩ ∈ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))
6513, 42, 46, 49, 52, 53, 64cofu2 17936 . . 3 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd𝐻)((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)))
666, 65eqtrid 2786 . 2 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = ((((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd𝐻)((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)))
6714, 13, 53, 39, 41, 49prf1 18255 . . . . . 6 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩)
6813, 28, 38, 49cofu1 17934 . . . . . . . 8 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)))
69 fvex 6919 . . . . . . . . . . 11 (1st𝑌) ∈ V
70 fvex 6919 . . . . . . . . . . . 12 (2nd𝑌) ∈ V
7170tposex 8283 . . . . . . . . . . 11 tpos (2nd𝑌) ∈ V
7269, 71op1st 8020 . . . . . . . . . 10 (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌)
7372a1i 11 . . . . . . . . 9 (𝜑 → (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌))
747, 13, 53, 26, 18, 27, 492ndf1 18250 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
75 op2ndg 8025 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
7647, 48, 75syl2anc 584 . . . . . . . . . 10 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
7774, 76eqtrd 2774 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
7873, 77fveq12d 6913 . . . . . . . 8 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st𝑌)‘𝑋))
7968, 78eqtrd 2774 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st𝑌)‘𝑋))
807, 13, 53, 26, 18, 40, 491stf1 18247 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
81 op1stg 8024 . . . . . . . . 9 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
8247, 48, 81syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
8380, 82eqtrd 2774 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
8479, 83opeq12d 4885 . . . . . 6 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
8567, 84eqtrd 2774 . . . . 5 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
8614, 13, 53, 39, 41, 52prf1 18255 . . . . . 6 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩) = ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩)⟩)
8713, 28, 38, 52cofu1 17934 . . . . . . . 8 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩) = ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)))
887, 13, 53, 26, 18, 27, 522ndf1 18250 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩) = (2nd ‘⟨𝐺, 𝑃⟩))
89 op2ndg 8025 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
9050, 51, 89syl2anc 584 . . . . . . . . . 10 (𝜑 → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
9188, 90eqtrd 2774 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩) = 𝑃)
9273, 91fveq12d 6913 . . . . . . . 8 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = ((1st𝑌)‘𝑃))
9387, 92eqtrd 2774 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩) = ((1st𝑌)‘𝑃))
947, 13, 53, 26, 18, 40, 521stf1 18247 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = (1st ‘⟨𝐺, 𝑃⟩))
95 op1stg 8024 . . . . . . . . 9 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
9650, 51, 95syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
9794, 96eqtrd 2774 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = 𝐺)
9893, 97opeq12d 4885 . . . . . 6 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩)⟩ = ⟨((1st𝑌)‘𝑃), 𝐺⟩)
9986, 98eqtrd 2774 . . . . 5 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩) = ⟨((1st𝑌)‘𝑃), 𝐺⟩)
10085, 99oveq12d 7448 . . . 4 (𝜑 → (((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd𝐻)((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩)) = (⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩))
10114, 13, 53, 39, 41, 49, 52, 64prf2 18257 . . . . 5 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ⟨((⟨𝐹, 𝑋⟩(2nd ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩), ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)⟩)
10213, 28, 38, 49, 52, 53, 64cofu2 17936 . . . . . . 7 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)))
10369, 71op2nd 8021 . . . . . . . . . . 11 (2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = tpos (2nd𝑌)
104103oveqi 7443 . . . . . . . . . 10 (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)tpos (2nd𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩))
105 ovtpos 8264 . . . . . . . . . 10 (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)tpos (2nd𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)(2nd𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩))
106104, 105eqtri 2762 . . . . . . . . 9 (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)(2nd𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩))
10791, 77oveq12d 7448 . . . . . . . . 9 (𝜑 → (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)(2nd𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = (𝑃(2nd𝑌)𝑋))
108106, 107eqtrid 2786 . . . . . . . 8 (𝜑 → (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = (𝑃(2nd𝑌)𝑋))
1097, 13, 53, 26, 18, 27, 49, 522ndf2 18251 . . . . . . . . . 10 (𝜑 → (⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩) = (2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)))
110109fveq1d 6908 . . . . . . . . 9 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩))
11164fvresd 6926 . . . . . . . . 9 (𝜑 → ((2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (2nd ‘⟨𝐴, 𝐾⟩))
112 op2ndg 8025 . . . . . . . . . 10 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋)) → (2nd ‘⟨𝐴, 𝐾⟩) = 𝐾)
11354, 55, 112syl2anc 584 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐴, 𝐾⟩) = 𝐾)
114110, 111, 1133eqtrd 2778 . . . . . . . 8 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = 𝐾)
115108, 114fveq12d 6913 . . . . . . 7 (𝜑 → ((((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)) = ((𝑃(2nd𝑌)𝑋)‘𝐾))
116102, 115eqtrd 2774 . . . . . 6 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((𝑃(2nd𝑌)𝑋)‘𝐾))
1177, 13, 53, 26, 18, 40, 49, 521stf2 18248 . . . . . . . 8 (𝜑 → (⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩) = (1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)))
118117fveq1d 6908 . . . . . . 7 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩))
11964fvresd 6926 . . . . . . 7 (𝜑 → ((1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (1st ‘⟨𝐴, 𝐾⟩))
120 op1stg 8024 . . . . . . . 8 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋)) → (1st ‘⟨𝐴, 𝐾⟩) = 𝐴)
12154, 55, 120syl2anc 584 . . . . . . 7 (𝜑 → (1st ‘⟨𝐴, 𝐾⟩) = 𝐴)
122118, 119, 1213eqtrd 2778 . . . . . 6 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = 𝐴)
123116, 122opeq12d 4885 . . . . 5 (𝜑 → ⟨((⟨𝐹, 𝑋⟩(2nd ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩), ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)⟩ = ⟨((𝑃(2nd𝑌)𝑋)‘𝐾), 𝐴⟩)
124101, 123eqtrd 2774 . . . 4 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ⟨((𝑃(2nd𝑌)𝑋)‘𝐾), 𝐴⟩)
125100, 124fveq12d 6913 . . 3 (𝜑 → ((((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd𝐻)((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)) = ((⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)‘⟨((𝑃(2nd𝑌)𝑋)‘𝐾), 𝐴⟩))
126 df-ov 7433 . . 3 (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴) = ((⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)‘⟨((𝑃(2nd𝑌)𝑋)‘𝐾), 𝐴⟩)
127125, 126eqtr4di 2792 . 2 (𝜑 → ((((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd𝐻)((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))
12866, 127eqtrd 2774 1 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  cun 3960  wss 3962  cop 4636   class class class wbr 5147   × cxp 5686  ran crn 5689  cres 5690  Rel wrel 5693  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  tpos ctpos 8248  Basecbs 17244  Hom chom 17308  Catccat 17708  Idccid 17709  Homf chomf 17710  oppCatcoppc 17755   Func cfunc 17904  func ccofu 17906   Nat cnat 17995   FuncCat cfuc 17996  SetCatcsetc 18128   ×c cxpc 18223   1stF c1stf 18224   2ndF c2ndf 18225   ⟨,⟩F cprf 18226   evalF cevlf 18265  HomFchof 18304  Yoncyon 18305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-hom 17321  df-cco 17322  df-cat 17712  df-cid 17713  df-homf 17714  df-comf 17715  df-oppc 17756  df-func 17908  df-cofu 17910  df-nat 17997  df-fuc 17998  df-setc 18129  df-xpc 18227  df-1stf 18228  df-2ndf 18229  df-prf 18230  df-curf 18270  df-hof 18306  df-yon 18307
This theorem is referenced by:  yonedalem3b  18335
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