MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  yonedalem22 Structured version   Visualization version   GIF version

Theorem yonedalem22 18323
Description: Lemma for yoneda 18328. (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 7444 . . . . 5 (⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩) = (⟨𝐹, 𝑋⟩(2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)
43oveqi 7444 . . . 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 7434 . . . 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 2765 . . 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 2737 . . . . 5 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
8 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
98fucbas 18008 . . . . 5 (𝑂 Func 𝑆) = (Base‘𝑄)
10 yoneda.o . . . . . 6 𝑂 = (oppCat‘𝐶)
11 yoneda.b . . . . . 6 𝐵 = (Base‘𝐶)
1210, 11oppcbas 17761 . . . . 5 𝐵 = (Base‘𝑂)
137, 9, 12xpcbas 18223 . . . 4 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
14 eqid 2737 . . . . 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 2737 . . . . 5 ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
16 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
1710oppccat 17765 . . . . . . . . 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 4194 . . . . . . . . . 10 (𝜑𝑈𝑉)
2219, 21ssexd 5324 . . . . . . . . 9 (𝜑𝑈 ∈ V)
23 yoneda.s . . . . . . . . . 10 𝑆 = (SetCat‘𝑈)
2423setccat 18130 . . . . . . . . 9 (𝑈 ∈ V → 𝑆 ∈ Cat)
2522, 24syl 17 . . . . . . . 8 (𝜑𝑆 ∈ Cat)
268, 18, 25fuccat 18018 . . . . . . 7 (𝜑𝑄 ∈ Cat)
27 eqid 2737 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
287, 26, 18, 272ndfcl 18243 . . . . . 6 (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
29 eqid 2737 . . . . . . . 8 (oppCat‘𝑄) = (oppCat‘𝑄)
30 relfunc 17907 . . . . . . . . 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 18307 . . . . . . . . 9 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
34 1st2ndbr 8067 . . . . . . . . 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 17920 . . . . . . 7 (𝜑 → (1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌))
37 df-br 5144 . . . . . . 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 17933 . . . . 5 (𝜑 → (⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
40 eqid 2737 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
417, 26, 18, 401stfcl 18242 . . . . 5 (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
4214, 15, 39, 41prfcl 18248 . . . 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 4193 . . . . 5 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
4643, 29, 44, 26, 19, 45hofcl 18304 . . . 4 (𝜑𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
47 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
48 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
4947, 48opelxpd 5724 . . . 4 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
50 yonedalem22.g . . . . 5 (𝜑𝐺 ∈ (𝑂 Func 𝑆))
51 yonedalem22.p . . . . 5 (𝜑𝑃𝐵)
5250, 51opelxpd 5724 . . . 4 (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
53 eqid 2737 . . . 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 2737 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
5756, 10oppchom 17758 . . . . . . 7 (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
5855, 57eleqtrrdi 2852 . . . . . 6 (𝜑𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
5954, 58opelxpd 5724 . . . . 5 (𝜑 → ⟨𝐴, 𝐾⟩ ∈ ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
60 eqid 2737 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
618, 60fuchom 18009 . . . . . 6 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
62 eqid 2737 . . . . . 6 (Hom ‘𝑂) = (Hom ‘𝑂)
637, 9, 12, 61, 62, 47, 48, 50, 51, 53xpchom2 18231 . . . . 5 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
6459, 63eleqtrrd 2844 . . . 4 (𝜑 → ⟨𝐴, 𝐾⟩ ∈ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))
6513, 42, 46, 49, 52, 53, 64cofu2 17931 . . 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 2789 . 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 18245 . . . . . 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 17929 . . . . . . . 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 8285 . . . . . . . . . . 11 tpos (2nd𝑌) ∈ V
7269, 71op1st 8022 . . . . . . . . . 10 (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌)
7372a1i 11 . . . . . . . . 9 (𝜑 → (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌))
747, 13, 53, 26, 18, 27, 492ndf1 18240 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
75 op2ndg 8027 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
7647, 48, 75syl2anc 584 . . . . . . . . . 10 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
7774, 76eqtrd 2777 . . . . . . . . 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 2777 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st𝑌)‘𝑋))
807, 13, 53, 26, 18, 40, 491stf1 18237 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
81 op1stg 8026 . . . . . . . . 9 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
8247, 48, 81syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
8380, 82eqtrd 2777 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
8479, 83opeq12d 4881 . . . . . 6 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
8567, 84eqtrd 2777 . . . . 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 18245 . . . . . 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 17929 . . . . . . . 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 18240 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩) = (2nd ‘⟨𝐺, 𝑃⟩))
89 op2ndg 8027 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
9050, 51, 89syl2anc 584 . . . . . . . . . 10 (𝜑 → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
9188, 90eqtrd 2777 . . . . . . . . 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 2777 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩) = ((1st𝑌)‘𝑃))
947, 13, 53, 26, 18, 40, 521stf1 18237 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = (1st ‘⟨𝐺, 𝑃⟩))
95 op1stg 8026 . . . . . . . . 9 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
9650, 51, 95syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
9794, 96eqtrd 2777 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = 𝐺)
9893, 97opeq12d 4881 . . . . . 6 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩)⟩ = ⟨((1st𝑌)‘𝑃), 𝐺⟩)
9986, 98eqtrd 2777 . . . . 5 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩) = ⟨((1st𝑌)‘𝑃), 𝐺⟩)
10085, 99oveq12d 7449 . . . 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 18247 . . . . 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 17931 . . . . . . 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 8023 . . . . . . . . . . 11 (2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = tpos (2nd𝑌)
104103oveqi 7444 . . . . . . . . . 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 8266 . . . . . . . . . 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 2765 . . . . . . . . 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 7449 . . . . . . . . 9 (𝜑 → (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)(2nd𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = (𝑃(2nd𝑌)𝑋))
108106, 107eqtrid 2789 . . . . . . . 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 18241 . . . . . . . . . 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 8027 . . . . . . . . . 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 2781 . . . . . . . 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 2777 . . . . . 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 18238 . . . . . . . 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 8026 . . . . . . . 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 2781 . . . . . 6 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = 𝐴)
123116, 122opeq12d 4881 . . . . 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 2777 . . . 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 7434 . . 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 2795 . 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 2777 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 1540  wcel 2108  Vcvv 3480  cun 3949  wss 3951  cop 4632   class class class wbr 5143   × cxp 5683  ran crn 5686  cres 5687  Rel wrel 5690  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  tpos ctpos 8250  Basecbs 17247  Hom chom 17308  Catccat 17707  Idccid 17708  Homf chomf 17709  oppCatcoppc 17754   Func cfunc 17899  func ccofu 17901   Nat cnat 17989   FuncCat cfuc 17990  SetCatcsetc 18120   ×c cxpc 18213   1stF c1stf 18214   2ndF c2ndf 18215   ⟨,⟩F cprf 18216   evalF cevlf 18254  HomFchof 18293  Yoncyon 18294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-homf 17713  df-comf 17714  df-oppc 17755  df-func 17903  df-cofu 17905  df-nat 17991  df-fuc 17992  df-setc 18121  df-xpc 18217  df-1stf 18218  df-2ndf 18219  df-prf 18220  df-curf 18259  df-hof 18295  df-yon 18296
This theorem is referenced by:  yonedalem3b  18324
  Copyright terms: Public domain W3C validator