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Theorem yonedalem22 18167
Description: Lemma for yoneda 18172. (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 6845 . . . . . 6 (2nd𝑍) = (2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
32oveqi 7370 . . . . 5 (⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩) = (⟨𝐹, 𝑋⟩(2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)
43oveqi 7370 . . . 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 7360 . . . 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 2764 . . 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 2736 . . . . 5 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
8 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
98fucbas 17848 . . . . 5 (𝑂 Func 𝑆) = (Base‘𝑄)
10 yoneda.o . . . . . 6 𝑂 = (oppCat‘𝐶)
11 yoneda.b . . . . . 6 𝐵 = (Base‘𝐶)
1210, 11oppcbas 17599 . . . . 5 𝐵 = (Base‘𝑂)
137, 9, 12xpcbas 18066 . . . 4 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
14 eqid 2736 . . . . 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 2736 . . . . 5 ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
16 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
1710oppccat 17604 . . . . . . . . 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 4148 . . . . . . . . . 10 (𝜑𝑈𝑉)
2219, 21ssexd 5281 . . . . . . . . 9 (𝜑𝑈 ∈ V)
23 yoneda.s . . . . . . . . . 10 𝑆 = (SetCat‘𝑈)
2423setccat 17971 . . . . . . . . 9 (𝑈 ∈ V → 𝑆 ∈ Cat)
2522, 24syl 17 . . . . . . . 8 (𝜑𝑆 ∈ Cat)
268, 18, 25fuccat 17859 . . . . . . 7 (𝜑𝑄 ∈ Cat)
27 eqid 2736 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
287, 26, 18, 272ndfcl 18086 . . . . . 6 (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
29 eqid 2736 . . . . . . . 8 (oppCat‘𝑄) = (oppCat‘𝑄)
30 relfunc 17748 . . . . . . . . 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 18151 . . . . . . . . 9 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
34 1st2ndbr 7974 . . . . . . . . 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 17761 . . . . . . 7 (𝜑 → (1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌))
37 df-br 5106 . . . . . . 7 ((1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌) ↔ ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3836, 37sylib 217 . . . . . 6 (𝜑 → ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3928, 38cofucl 17774 . . . . 5 (𝜑 → (⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
40 eqid 2736 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
417, 26, 18, 401stfcl 18085 . . . . 5 (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
4214, 15, 39, 41prfcl 18091 . . . 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 4147 . . . . 5 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
4643, 29, 44, 26, 19, 45hofcl 18148 . . . 4 (𝜑𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
47 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
48 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
4947, 48opelxpd 5671 . . . 4 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
50 yonedalem22.g . . . . 5 (𝜑𝐺 ∈ (𝑂 Func 𝑆))
51 yonedalem22.p . . . . 5 (𝜑𝑃𝐵)
5250, 51opelxpd 5671 . . . 4 (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
53 eqid 2736 . . . 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 2736 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
5756, 10oppchom 17596 . . . . . . 7 (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
5855, 57eleqtrrdi 2849 . . . . . 6 (𝜑𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
5954, 58opelxpd 5671 . . . . 5 (𝜑 → ⟨𝐴, 𝐾⟩ ∈ ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
60 eqid 2736 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
618, 60fuchom 17849 . . . . . 6 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
62 eqid 2736 . . . . . 6 (Hom ‘𝑂) = (Hom ‘𝑂)
637, 9, 12, 61, 62, 47, 48, 50, 51, 53xpchom2 18074 . . . . 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 17772 . . 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 2788 . 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 18088 . . . . . 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 17770 . . . . . . . 8 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)))
69 fvex 6855 . . . . . . . . . . 11 (1st𝑌) ∈ V
70 fvex 6855 . . . . . . . . . . . 12 (2nd𝑌) ∈ V
7170tposex 8191 . . . . . . . . . . 11 tpos (2nd𝑌) ∈ V
7269, 71op1st 7929 . . . . . . . . . 10 (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌)
7372a1i 11 . . . . . . . . 9 (𝜑 → (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌))
747, 13, 53, 26, 18, 27, 492ndf1 18083 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
75 op2ndg 7934 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
7647, 48, 75syl2anc 584 . . . . . . . . . 10 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
7774, 76eqtrd 2776 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
7873, 77fveq12d 6849 . . . . . . . 8 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st𝑌)‘𝑋))
7968, 78eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st𝑌)‘𝑋))
807, 13, 53, 26, 18, 40, 491stf1 18080 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
81 op1stg 7933 . . . . . . . . 9 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
8247, 48, 81syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
8380, 82eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
8479, 83opeq12d 4838 . . . . . 6 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
8567, 84eqtrd 2776 . . . . 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 18088 . . . . . 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 17770 . . . . . . . 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 18083 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩) = (2nd ‘⟨𝐺, 𝑃⟩))
89 op2ndg 7934 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
9050, 51, 89syl2anc 584 . . . . . . . . . 10 (𝜑 → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
9188, 90eqtrd 2776 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩) = 𝑃)
9273, 91fveq12d 6849 . . . . . . . 8 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = ((1st𝑌)‘𝑃))
9387, 92eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩) = ((1st𝑌)‘𝑃))
947, 13, 53, 26, 18, 40, 521stf1 18080 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = (1st ‘⟨𝐺, 𝑃⟩))
95 op1stg 7933 . . . . . . . . 9 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
9650, 51, 95syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
9794, 96eqtrd 2776 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = 𝐺)
9893, 97opeq12d 4838 . . . . . 6 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩)⟩ = ⟨((1st𝑌)‘𝑃), 𝐺⟩)
9986, 98eqtrd 2776 . . . . 5 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩) = ⟨((1st𝑌)‘𝑃), 𝐺⟩)
10085, 99oveq12d 7375 . . . 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 18090 . . . . 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 17772 . . . . . . 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 7930 . . . . . . . . . . 11 (2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = tpos (2nd𝑌)
104103oveqi 7370 . . . . . . . . . 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 8172 . . . . . . . . . 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 2764 . . . . . . . . 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 7375 . . . . . . . . 9 (𝜑 → (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)(2nd𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = (𝑃(2nd𝑌)𝑋))
108106, 107eqtrid 2788 . . . . . . . 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 18084 . . . . . . . . . 10 (𝜑 → (⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩) = (2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)))
110109fveq1d 6844 . . . . . . . . 9 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩))
11164fvresd 6862 . . . . . . . . 9 (𝜑 → ((2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (2nd ‘⟨𝐴, 𝐾⟩))
112 op2ndg 7934 . . . . . . . . . 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 2780 . . . . . . . 8 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = 𝐾)
115108, 114fveq12d 6849 . . . . . . 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 2776 . . . . . 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 18081 . . . . . . . 8 (𝜑 → (⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩) = (1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)))
118117fveq1d 6844 . . . . . . 7 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩))
11964fvresd 6862 . . . . . . 7 (𝜑 → ((1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (1st ‘⟨𝐴, 𝐾⟩))
120 op1stg 7933 . . . . . . . 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 2780 . . . . . 6 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = 𝐴)
123116, 122opeq12d 4838 . . . . 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 2776 . . . 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 6849 . . 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 7360 . . 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 2794 . 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 2776 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 1541  wcel 2106  Vcvv 3445  cun 3908  wss 3910  cop 4592   class class class wbr 5105   × cxp 5631  ran crn 5634  cres 5635  Rel wrel 5638  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  tpos ctpos 8156  Basecbs 17083  Hom chom 17144  Catccat 17544  Idccid 17545  Homf chomf 17546  oppCatcoppc 17591   Func cfunc 17740  func ccofu 17742   Nat cnat 17828   FuncCat cfuc 17829  SetCatcsetc 17961   ×c cxpc 18056   1stF c1stf 18057   2ndF c2ndf 18058   ⟨,⟩F cprf 18059   evalF cevlf 18098  HomFchof 18137  Yoncyon 18138
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-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-tpos 8157  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-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-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-homf 17550  df-comf 17551  df-oppc 17592  df-func 17744  df-cofu 17746  df-nat 17830  df-fuc 17831  df-setc 17962  df-xpc 18060  df-1stf 18061  df-2ndf 18062  df-prf 18063  df-curf 18103  df-hof 18139  df-yon 18140
This theorem is referenced by:  yonedalem3b  18168
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