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Theorem yonedalem22 16839
Description: Lemma for yoneda 16844. (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 6151 . . . . . 6 (2nd𝑍) = (2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
32oveqi 6617 . . . . 5 (⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩) = (⟨𝐹, 𝑋⟩(2nd ‘(𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)
43oveqi 6617 . . . 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 6607 . . . 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 2643 . . 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 2621 . . . . 5 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
8 yoneda.q . . . . . 6 𝑄 = (𝑂 FuncCat 𝑆)
98fucbas 16541 . . . . 5 (𝑂 Func 𝑆) = (Base‘𝑄)
10 yoneda.o . . . . . 6 𝑂 = (oppCat‘𝐶)
11 yoneda.b . . . . . 6 𝐵 = (Base‘𝐶)
1210, 11oppcbas 16299 . . . . 5 𝐵 = (Base‘𝑂)
137, 9, 12xpcbas 16739 . . . 4 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
14 eqid 2621 . . . . 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 2621 . . . . 5 ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
16 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
1710oppccat 16303 . . . . . . . . 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 3769 . . . . . . . . . 10 (𝜑𝑈𝑉)
2219, 21ssexd 4765 . . . . . . . . 9 (𝜑𝑈 ∈ V)
23 yoneda.s . . . . . . . . . 10 𝑆 = (SetCat‘𝑈)
2423setccat 16656 . . . . . . . . 9 (𝑈 ∈ V → 𝑆 ∈ Cat)
2522, 24syl 17 . . . . . . . 8 (𝜑𝑆 ∈ Cat)
268, 18, 25fuccat 16551 . . . . . . 7 (𝜑𝑄 ∈ Cat)
27 eqid 2621 . . . . . . 7 (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
287, 26, 18, 272ndfcl 16759 . . . . . 6 (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
29 eqid 2621 . . . . . . . 8 (oppCat‘𝑄) = (oppCat‘𝑄)
30 relfunc 16443 . . . . . . . . 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 16823 . . . . . . . . 9 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
34 1st2ndbr 7162 . . . . . . . . 9 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3530, 33, 34sylancr 694 . . . . . . . 8 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
3610, 29, 35funcoppc 16456 . . . . . . 7 (𝜑 → (1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌))
37 df-br 4614 . . . . . . 7 ((1st𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd𝑌) ↔ ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3836, 37sylib 208 . . . . . 6 (𝜑 → ⟨(1st𝑌), tpos (2nd𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
3928, 38cofucl 16469 . . . . 5 (𝜑 → (⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
40 eqid 2621 . . . . . 6 (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
417, 26, 18, 401stfcl 16758 . . . . 5 (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
4214, 15, 39, 41prfcl 16764 . . . 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 3768 . . . . 5 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
4643, 29, 44, 26, 19, 45hofcl 16820 . . . 4 (𝜑𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
47 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
48 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
49 opelxpi 5108 . . . . 5 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
5047, 48, 49syl2anc 692 . . . 4 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
51 yonedalem22.g . . . . 5 (𝜑𝐺 ∈ (𝑂 Func 𝑆))
52 yonedalem22.p . . . . 5 (𝜑𝑃𝐵)
53 opelxpi 5108 . . . . 5 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
5451, 52, 53syl2anc 692 . . . 4 (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
55 eqid 2621 . . . 4 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
56 yonedalem22.a . . . . . 6 (𝜑𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
57 yonedalem22.k . . . . . . 7 (𝜑𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))
58 eqid 2621 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
5958, 10oppchom 16296 . . . . . . 7 (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
6057, 59syl6eleqr 2709 . . . . . 6 (𝜑𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
61 opelxpi 5108 . . . . . 6 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃)) → ⟨𝐴, 𝐾⟩ ∈ ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
6256, 60, 61syl2anc 692 . . . . 5 (𝜑 → ⟨𝐴, 𝐾⟩ ∈ ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
63 eqid 2621 . . . . . . 7 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
648, 63fuchom 16542 . . . . . 6 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
65 eqid 2621 . . . . . 6 (Hom ‘𝑂) = (Hom ‘𝑂)
667, 9, 12, 64, 65, 47, 48, 51, 52, 55xpchom2 16747 . . . . 5 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
6762, 66eleqtrrd 2701 . . . 4 (𝜑 → ⟨𝐴, 𝐾⟩ ∈ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))
6813, 42, 46, 50, 54, 55, 67cofu2 16467 . . 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⟩)))
696, 68syl5eq 2667 . 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⟩)))
7014, 13, 55, 39, 41, 50prf1 16761 . . . . . 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⟩)⟩)
7113, 28, 38, 50cofu1 16465 . . . . . . . 8 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)))
72 fvex 6158 . . . . . . . . . . 11 (1st𝑌) ∈ V
73 fvex 6158 . . . . . . . . . . . 12 (2nd𝑌) ∈ V
7473tposex 7331 . . . . . . . . . . 11 tpos (2nd𝑌) ∈ V
7572, 74op1st 7121 . . . . . . . . . 10 (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌)
7675a1i 11 . . . . . . . . 9 (𝜑 → (1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = (1st𝑌))
777, 13, 55, 26, 18, 27, 502ndf1 16756 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
78 op2ndg 7126 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
7947, 48, 78syl2anc 692 . . . . . . . . . 10 (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
8077, 79eqtrd 2655 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
8176, 80fveq12d 6154 . . . . . . . 8 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st𝑌)‘𝑋))
8271, 81eqtrd 2655 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st𝑌)‘𝑋))
837, 13, 55, 26, 18, 40, 501stf1 16753 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
84 op1stg 7125 . . . . . . . . 9 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
8547, 48, 84syl2anc 692 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
8683, 85eqtrd 2655 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
8782, 86opeq12d 4378 . . . . . 6 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
8870, 87eqtrd 2655 . . . . 5 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st𝑌)‘𝑋), 𝐹⟩)
8914, 13, 55, 39, 41, 54prf1 16761 . . . . . 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⟩)⟩)
9013, 28, 38, 54cofu1 16465 . . . . . . . 8 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩) = ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)))
917, 13, 55, 26, 18, 27, 542ndf1 16756 . . . . . . . . . 10 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩) = (2nd ‘⟨𝐺, 𝑃⟩))
92 op2ndg 7126 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
9351, 52, 92syl2anc 692 . . . . . . . . . 10 (𝜑 → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
9491, 93eqtrd 2655 . . . . . . . . 9 (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩) = 𝑃)
9576, 94fveq12d 6154 . . . . . . . 8 (𝜑 → ((1st ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = ((1st𝑌)‘𝑃))
9690, 95eqtrd 2655 . . . . . . 7 (𝜑 → ((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩) = ((1st𝑌)‘𝑃))
977, 13, 55, 26, 18, 40, 541stf1 16753 . . . . . . . 8 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = (1st ‘⟨𝐺, 𝑃⟩))
98 op1stg 7125 . . . . . . . . 9 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
9951, 52, 98syl2anc 692 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
10097, 99eqtrd 2655 . . . . . . 7 (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = 𝐺)
10196, 100opeq12d 4378 . . . . . 6 (𝜑 → ⟨((1st ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩)⟩ = ⟨((1st𝑌)‘𝑃), 𝐺⟩)
10289, 101eqtrd 2655 . . . . 5 (𝜑 → ((1st ‘((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩) = ⟨((1st𝑌)‘𝑃), 𝐺⟩)
10388, 102oveq12d 6622 . . . 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⟩))
10414, 13, 55, 39, 41, 50, 54, 67prf2 16763 . . . . 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⟩)⟩)
10513, 28, 38, 50, 54, 55, 67cofu2 16467 . . . . . . 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⟩)))
10672, 74op2nd 7122 . . . . . . . . . . 11 (2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩) = tpos (2nd𝑌)
107106oveqi 6617 . . . . . . . . . 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⟩))
108 ovtpos 7312 . . . . . . . . . 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⟩))
109107, 108eqtri 2643 . . . . . . . . 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⟩))
11094, 80oveq12d 6622 . . . . . . . . 9 (𝜑 → (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)(2nd𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = (𝑃(2nd𝑌)𝑋))
111109, 110syl5eq 2667 . . . . . . . 8 (𝜑 → (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st𝑌), tpos (2nd𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = (𝑃(2nd𝑌)𝑋))
1127, 13, 55, 26, 18, 27, 50, 542ndf2 16757 . . . . . . . . . 10 (𝜑 → (⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩) = (2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)))
113112fveq1d 6150 . . . . . . . . 9 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩))
114 fvres 6164 . . . . . . . . . 10 (⟨𝐴, 𝐾⟩ ∈ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) → ((2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (2nd ‘⟨𝐴, 𝐾⟩))
11567, 114syl 17 . . . . . . . . 9 (𝜑 → ((2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (2nd ‘⟨𝐴, 𝐾⟩))
116 op2ndg 7126 . . . . . . . . . 10 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋)) → (2nd ‘⟨𝐴, 𝐾⟩) = 𝐾)
11756, 57, 116syl2anc 692 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝐴, 𝐾⟩) = 𝐾)
118113, 115, 1173eqtrd 2659 . . . . . . . 8 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = 𝐾)
119111, 118fveq12d 6154 . . . . . . 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))
120105, 119eqtrd 2655 . . . . . 6 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((𝑃(2nd𝑌)𝑋)‘𝐾))
1217, 13, 55, 26, 18, 40, 50, 541stf2 16754 . . . . . . . 8 (𝜑 → (⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩) = (1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)))
122121fveq1d 6150 . . . . . . 7 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩))
123 fvres 6164 . . . . . . . 8 (⟨𝐴, 𝐾⟩ ∈ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) → ((1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (1st ‘⟨𝐴, 𝐾⟩))
12467, 123syl 17 . . . . . . 7 (𝜑 → ((1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (1st ‘⟨𝐴, 𝐾⟩))
125 op1stg 7125 . . . . . . . 8 ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋)) → (1st ‘⟨𝐴, 𝐾⟩) = 𝐴)
12656, 57, 125syl2anc 692 . . . . . . 7 (𝜑 → (1st ‘⟨𝐴, 𝐾⟩) = 𝐴)
127122, 124, 1263eqtrd 2659 . . . . . 6 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = 𝐴)
128120, 127opeq12d 4378 . . . . 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⟩)
129104, 128eqtrd 2655 . . . 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⟩)
130103, 129fveq12d 6154 . . 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⟩))
131 df-ov 6607 . . 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⟩)
132130, 131syl6eqr 2673 . 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))
13369, 132eqtrd 2655 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 1480  wcel 1987  Vcvv 3186  cun 3553  wss 3555  cop 4154   class class class wbr 4613   × cxp 5072  ran crn 5075  cres 5076  Rel wrel 5079  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  tpos ctpos 7296  Basecbs 15781  Hom chom 15873  Catccat 16246  Idccid 16247  Homf chomf 16248  oppCatcoppc 16292   Func cfunc 16435  func ccofu 16437   Nat cnat 16522   FuncCat cfuc 16523  SetCatcsetc 16646   ×c cxpc 16729   1stF c1stf 16730   2ndF c2ndf 16731   ⟨,⟩F cprf 16732   evalF cevlf 16770  HomFchof 16809  Yoncyon 16810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-hom 15887  df-cco 15888  df-cat 16250  df-cid 16251  df-homf 16252  df-comf 16253  df-oppc 16293  df-func 16439  df-cofu 16441  df-nat 16524  df-fuc 16525  df-setc 16647  df-xpc 16733  df-1stf 16734  df-2ndf 16735  df-prf 16736  df-curf 16775  df-hof 16811  df-yon 16812
This theorem is referenced by:  yonedalem3b  16840
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