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Theorem yonedalem3b 17358
Description: Lemma for yoneda 17362. (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))
yonedalem3.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
Assertion
Ref Expression
yonedalem3b (𝜑 → ((𝐺𝑀𝑃)(⟨(𝐹(1st𝑍)𝑋), (𝐺(1st𝑍)𝑃)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾)(⟨(𝐹(1st𝑍)𝑋), (𝐹(1st𝐸)𝑋)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐹𝑀𝑋)))
Distinct variable groups:   𝑓,𝑎,𝑥, 1   𝐴,𝑎   𝐶,𝑎,𝑓,𝑥   𝐸,𝑎,𝑓   𝐹,𝑎,𝑓,𝑥   𝐾,𝑎   𝐵,𝑎,𝑓,𝑥   𝐺,𝑎,𝑓,𝑥   𝑂,𝑎,𝑓,𝑥   𝑆,𝑎,𝑓,𝑥   𝑄,𝑎,𝑓,𝑥   𝑇,𝑓   𝑃,𝑎,𝑓,𝑥   𝜑,𝑎,𝑓,𝑥   𝑌,𝑎,𝑓,𝑥   𝑍,𝑎,𝑓,𝑥   𝑋,𝑎,𝑓,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑓,𝑎)   𝐸(𝑥)   𝐻(𝑥,𝑓,𝑎)   𝐾(𝑥,𝑓)   𝑀(𝑥,𝑓,𝑎)   𝑉(𝑥,𝑓,𝑎)   𝑊(𝑥,𝑓,𝑎)

Proof of Theorem yonedalem3b
Dummy variables 𝑏 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7024 . . . . . . . 8 (𝑏 = 𝑎 → (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏) = (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎))
21oveq1d 7031 . . . . . . 7 (𝑏 = 𝑎 → ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾)) = ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾)))
32fveq1d 6540 . . . . . 6 (𝑏 = 𝑎 → (((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃) = (((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃))
43fveq1d 6540 . . . . 5 (𝑏 = 𝑎 → ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃)) = ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃)))
54cbvmptv 5061 . . . 4 (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃)))
6 yoneda.q . . . . . . . . 9 𝑄 = (𝑂 FuncCat 𝑆)
7 eqid 2795 . . . . . . . . 9 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8 yoneda.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
9 yoneda.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
108, 9oppcbas 16817 . . . . . . . . 9 𝐵 = (Base‘𝑂)
11 eqid 2795 . . . . . . . . 9 (comp‘𝑆) = (comp‘𝑆)
12 eqid 2795 . . . . . . . . 9 (comp‘𝑄) = (comp‘𝑄)
13 eqid 2795 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
146, 7fuchom 17060 . . . . . . . . . . . 12 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
15 relfunc 16961 . . . . . . . . . . . . 13 Rel (𝐶 Func 𝑄)
16 yoneda.y . . . . . . . . . . . . . 14 𝑌 = (Yon‘𝐶)
17 yoneda.c . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ Cat)
18 yoneda.s . . . . . . . . . . . . . 14 𝑆 = (SetCat‘𝑈)
19 yoneda.w . . . . . . . . . . . . . . 15 (𝜑𝑉𝑊)
20 yoneda.v . . . . . . . . . . . . . . . 16 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2120unssbd 4085 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
2219, 21ssexd 5119 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ V)
23 yoneda.u . . . . . . . . . . . . . 14 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2416, 17, 8, 18, 6, 22, 23yoncl 17341 . . . . . . . . . . . . 13 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
25 1st2ndbr 7597 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
2615, 24, 25sylancr 587 . . . . . . . . . . . 12 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
27 yonedalem22.p . . . . . . . . . . . 12 (𝜑𝑃𝐵)
28 yonedalem21.x . . . . . . . . . . . 12 (𝜑𝑋𝐵)
299, 13, 14, 26, 27, 28funcf2 16967 . . . . . . . . . . 11 (𝜑 → (𝑃(2nd𝑌)𝑋):(𝑃(Hom ‘𝐶)𝑋)⟶(((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st𝑌)‘𝑋)))
30 yonedalem22.k . . . . . . . . . . 11 (𝜑𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))
3129, 30ffvelrnd 6717 . . . . . . . . . 10 (𝜑 → ((𝑃(2nd𝑌)𝑋)‘𝐾) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st𝑌)‘𝑋)))
3231adantr 481 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑃(2nd𝑌)𝑋)‘𝐾) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st𝑌)‘𝑋)))
33 simpr 485 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
34 yonedalem22.a . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
3534adantr 481 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
366, 7, 12, 33, 35fuccocl 17063 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐺))
3727adantr 481 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑃𝐵)
386, 7, 10, 11, 12, 32, 36, 37fuccoval 17062 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃) = (((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)‘𝑃)(⟨((1st ‘((1st𝑌)‘𝑃))‘𝑃), ((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)))
396, 7, 10, 11, 12, 33, 35, 37fuccoval 17062 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)‘𝑃) = ((𝐴𝑃)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑃), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(𝑎𝑃)))
4022adantr 481 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
41 eqid 2795 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
42 relfunc 16961 . . . . . . . . . . . . . . . 16 Rel (𝑂 Func 𝑆)
436fucbas 17059 . . . . . . . . . . . . . . . . . 18 (𝑂 Func 𝑆) = (Base‘𝑄)
449, 43, 26funcf1 16965 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
4544, 28ffvelrnd 6717 . . . . . . . . . . . . . . . 16 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
46 1st2ndbr 7597 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
4742, 45, 46sylancr 587 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
4810, 41, 47funcf1 16965 . . . . . . . . . . . . . 14 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4918, 22setcbas 17167 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝑆))
5049feq3d 6369 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
5148, 50mpbird 258 . . . . . . . . . . . . 13 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
5251, 27ffvelrnd 6717 . . . . . . . . . . . 12 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∈ 𝑈)
5352adantr 481 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∈ 𝑈)
54 yonedalem21.f . . . . . . . . . . . . . . . 16 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
55 1st2ndbr 7597 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
5642, 54, 55sylancr 587 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
5710, 41, 56funcf1 16965 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
5849feq3d 6369 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹):𝐵𝑈 ↔ (1st𝐹):𝐵⟶(Base‘𝑆)))
5957, 58mpbird 258 . . . . . . . . . . . . 13 (𝜑 → (1st𝐹):𝐵𝑈)
6059, 27ffvelrnd 6717 . . . . . . . . . . . 12 (𝜑 → ((1st𝐹)‘𝑃) ∈ 𝑈)
6160adantr 481 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑃) ∈ 𝑈)
62 yonedalem22.g . . . . . . . . . . . . . . . 16 (𝜑𝐺 ∈ (𝑂 Func 𝑆))
63 1st2ndbr 7597 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐺 ∈ (𝑂 Func 𝑆)) → (1st𝐺)(𝑂 Func 𝑆)(2nd𝐺))
6442, 62, 63sylancr 587 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝐺)(𝑂 Func 𝑆)(2nd𝐺))
6510, 41, 64funcf1 16965 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝑆))
6665, 27ffvelrnd 6717 . . . . . . . . . . . . 13 (𝜑 → ((1st𝐺)‘𝑃) ∈ (Base‘𝑆))
6766, 49eleqtrrd 2886 . . . . . . . . . . . 12 (𝜑 → ((1st𝐺)‘𝑃) ∈ 𝑈)
6867adantr 481 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐺)‘𝑃) ∈ 𝑈)
697, 33nat1st2nd 17050 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (⟨(1st ‘((1st𝑌)‘𝑋)), (2nd ‘((1st𝑌)‘𝑋))⟩(𝑂 Nat 𝑆)⟨(1st𝐹), (2nd𝐹)⟩))
70 eqid 2795 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
717, 69, 10, 70, 37natcl 17052 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑃) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑃)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
7218, 40, 70, 53, 61elsetchom 17170 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑃)(Hom ‘𝑆)((1st𝐹)‘𝑃)) ↔ (𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃)))
7371, 72mpbid 233 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃))
747, 34nat1st2nd 17050 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩(𝑂 Nat 𝑆)⟨(1st𝐺), (2nd𝐺)⟩))
757, 74, 10, 70, 27natcl 17052 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑃) ∈ (((1st𝐹)‘𝑃)(Hom ‘𝑆)((1st𝐺)‘𝑃)))
7618, 22, 70, 60, 67elsetchom 17170 . . . . . . . . . . . . 13 (𝜑 → ((𝐴𝑃) ∈ (((1st𝐹)‘𝑃)(Hom ‘𝑆)((1st𝐺)‘𝑃)) ↔ (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃)))
7775, 76mpbid 233 . . . . . . . . . . . 12 (𝜑 → (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃))
7877adantr 481 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃))
7918, 40, 11, 53, 61, 68, 73, 78setcco 17172 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑃), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(𝑎𝑃)) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8039, 79eqtrd 2831 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)‘𝑃) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8180oveq1d 7031 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)‘𝑃)(⟨((1st ‘((1st𝑌)‘𝑃))‘𝑃), ((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)) = (((𝐴𝑃) ∘ (𝑎𝑃))(⟨((1st ‘((1st𝑌)‘𝑃))‘𝑃), ((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)))
8244, 27ffvelrnd 6717 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝑌)‘𝑃) ∈ (𝑂 Func 𝑆))
83 1st2ndbr 7597 . . . . . . . . . . . . . 14 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑃) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑃)))
8442, 82, 83sylancr 587 . . . . . . . . . . . . 13 (𝜑 → (1st ‘((1st𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑃)))
8510, 41, 84funcf1 16965 . . . . . . . . . . . 12 (𝜑 → (1st ‘((1st𝑌)‘𝑃)):𝐵⟶(Base‘𝑆))
8685, 27ffvelrnd 6717 . . . . . . . . . . 11 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ (Base‘𝑆))
8786, 49eleqtrrd 2886 . . . . . . . . . 10 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
8887adantr 481 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
897, 31nat1st2nd 17050 . . . . . . . . . . . 12 (𝜑 → ((𝑃(2nd𝑌)𝑋)‘𝐾) ∈ (⟨(1st ‘((1st𝑌)‘𝑃)), (2nd ‘((1st𝑌)‘𝑃))⟩(𝑂 Nat 𝑆)⟨(1st ‘((1st𝑌)‘𝑋)), (2nd ‘((1st𝑌)‘𝑋))⟩))
907, 89, 10, 70, 27natcl 17052 . . . . . . . . . . 11 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃) ∈ (((1st ‘((1st𝑌)‘𝑃))‘𝑃)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
9118, 22, 70, 87, 52elsetchom 17170 . . . . . . . . . . 11 (𝜑 → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃) ∈ (((1st ‘((1st𝑌)‘𝑃))‘𝑃)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)) ↔ (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
9290, 91mpbid 233 . . . . . . . . . 10 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
9392adantr 481 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
94 fco 6399 . . . . . . . . . 10 (((𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃) ∧ (𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃)) → ((𝐴𝑃) ∘ (𝑎𝑃)):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐺)‘𝑃))
9578, 73, 94syl2anc 584 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃) ∘ (𝑎𝑃)):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐺)‘𝑃))
9618, 40, 11, 88, 53, 68, 93, 95setcco 17172 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴𝑃) ∘ (𝑎𝑃))(⟨((1st ‘((1st𝑌)‘𝑃))‘𝑃), ((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)) = (((𝐴𝑃) ∘ (𝑎𝑃)) ∘ (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)))
9738, 81, 963eqtrd 2835 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃) = (((𝐴𝑃) ∘ (𝑎𝑃)) ∘ (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)))
9897fveq1d 6540 . . . . . 6 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃)) = ((((𝐴𝑃) ∘ (𝑎𝑃)) ∘ (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃))‘( 1𝑃)))
99 yoneda.1 . . . . . . . . . 10 1 = (Id‘𝐶)
1009, 13, 99, 17, 27catidcl 16782 . . . . . . . . 9 (𝜑 → ( 1𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃))
10116, 9, 17, 27, 13, 27yon11 17343 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) = (𝑃(Hom ‘𝐶)𝑃))
102100, 101eleqtrrd 2886 . . . . . . . 8 (𝜑 → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
103102adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
104 fvco3 6627 . . . . . . 7 (((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∧ ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃)) → ((((𝐴𝑃) ∘ (𝑎𝑃)) ∘ (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃))‘( 1𝑃)) = (((𝐴𝑃) ∘ (𝑎𝑃))‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))))
10593, 103, 104syl2anc 584 . . . . . 6 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝐴𝑃) ∘ (𝑎𝑃)) ∘ (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃))‘( 1𝑃)) = (((𝐴𝑃) ∘ (𝑎𝑃))‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))))
10693, 103ffvelrnd 6717 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑃))
107 fvco3 6627 . . . . . . . 8 (((𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃) ∧ ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑃)) → (((𝐴𝑃) ∘ (𝑎𝑃))‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = ((𝐴𝑃)‘((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)))))
10873, 106, 107syl2anc 584 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴𝑃) ∘ (𝑎𝑃))‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = ((𝐴𝑃)‘((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)))))
10917adantr 481 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐶 ∈ Cat)
11028adantr 481 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋𝐵)
111 eqid 2795 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
11230adantr 481 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))
113100adantr 481 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃))
11416, 9, 109, 37, 13, 110, 111, 37, 112, 113yon2 17345 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) = (𝐾(⟨𝑃, 𝑃⟩(comp‘𝐶)𝑋)( 1𝑃)))
1159, 13, 99, 109, 37, 111, 110, 112catrid 16784 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐾(⟨𝑃, 𝑃⟩(comp‘𝐶)𝑋)( 1𝑃)) = 𝐾)
116114, 115eqtrd 2831 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) = 𝐾)
117116fveq2d 6542 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = ((𝑎𝑃)‘𝐾))
118 eqid 2795 . . . . . . . . . . . . . . 15 (Hom ‘𝑂) = (Hom ‘𝑂)
11910, 118, 70, 47, 28, 27funcf2 16967 . . . . . . . . . . . . . 14 (𝜑 → (𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12013, 8oppchom 16814 . . . . . . . . . . . . . . 15 (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
12130, 120syl6eleqr 2894 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
122119, 121ffvelrnd 6717 . . . . . . . . . . . . 13 (𝜑 → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12351, 28ffvelrnd 6717 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
12418, 22, 70, 123, 52elsetchom 17170 . . . . . . . . . . . . 13 (𝜑 → (((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)) ↔ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
125122, 124mpbid 233 . . . . . . . . . . . 12 (𝜑 → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
126125adantr 481 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
1279, 13, 99, 17, 28catidcl 16782 . . . . . . . . . . . . 13 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
12816, 9, 17, 28, 13, 28yon11 17343 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
129127, 128eleqtrrd 2886 . . . . . . . . . . . 12 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
130129adantr 481 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
131 fvco3 6627 . . . . . . . . . . 11 ((((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∧ ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋)) → (((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾))‘( 1𝑋)) = ((𝑎𝑃)‘(((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋))))
132126, 130, 131syl2anc 584 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾))‘( 1𝑋)) = ((𝑎𝑃)‘(((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋))))
133121adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
1347, 69, 10, 118, 11, 110, 37, 133nati 17054 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑋), ((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟩(comp‘𝑆)((1st𝐹)‘𝑃))((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)) = (((𝑋(2nd𝐹)𝑃)‘𝐾)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑋), ((1st𝐹)‘𝑋)⟩(comp‘𝑆)((1st𝐹)‘𝑃))(𝑎𝑋)))
135123adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
13618, 40, 11, 135, 53, 61, 126, 73setcco 17172 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑋), ((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟩(comp‘𝑆)((1st𝐹)‘𝑃))((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)) = ((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)))
13759, 28ffvelrnd 6717 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
138137adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
1397, 69, 10, 70, 110natcl 17052 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑋)))
14018, 40, 70, 135, 138elsetchom 17170 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑋) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑋)) ↔ (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋)))
141139, 140mpbid 233 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋))
14210, 118, 70, 56, 28, 27funcf2 16967 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋(2nd𝐹)𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
143142, 121ffvelrnd 6717 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑋(2nd𝐹)𝑃)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
14418, 22, 70, 137, 60elsetchom 17170 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑋(2nd𝐹)𝑃)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)) ↔ ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃)))
145143, 144mpbid 233 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃))
146145adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃))
14718, 40, 11, 135, 138, 61, 141, 146setcco 17172 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd𝐹)𝑃)‘𝐾)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑋), ((1st𝐹)‘𝑋)⟩(comp‘𝑆)((1st𝐹)‘𝑃))(𝑎𝑋)) = (((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋)))
148134, 136, 1473eqtr3d 2839 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)) = (((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋)))
149148fveq1d 6540 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾))‘( 1𝑋)) = ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)))
150127adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
15116, 9, 109, 110, 13, 110, 111, 37, 112, 150yon12 17344 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋)) = (( 1𝑋)(⟨𝑃, 𝑋⟩(comp‘𝐶)𝑋)𝐾))
1529, 13, 99, 109, 37, 111, 110, 112catlid 16783 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (( 1𝑋)(⟨𝑃, 𝑋⟩(comp‘𝐶)𝑋)𝐾) = 𝐾)
153151, 152eqtrd 2831 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋)) = 𝐾)
154153fveq2d 6542 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘(((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋))) = ((𝑎𝑃)‘𝐾))
155132, 149, 1543eqtr3d 2839 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = ((𝑎𝑃)‘𝐾))
156 fvco3 6627 . . . . . . . . . 10 (((𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋) ∧ ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
157141, 130, 156syl2anc 584 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
158117, 155, 1573eqtr2d 2837 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
159158fveq2d 6542 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃)‘((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)))) = ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋)))))
160108, 159eqtrd 2831 . . . . . 6 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝐴𝑃) ∘ (𝑎𝑃))‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋)))))
16198, 105, 1603eqtrd 2835 . . . . 5 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃)) = ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋)))))
162161mpteq2dva 5055 . . . 4 (𝜑 → (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
1635, 162syl5eq 2843 . . 3 (𝜑 → (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
164 eqid 2795 . . . . . . . . . . 11 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
165164, 43, 10xpcbas 17257 . . . . . . . . . 10 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
166 eqid 2795 . . . . . . . . . 10 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
167 eqid 2795 . . . . . . . . . 10 (Hom ‘𝑇) = (Hom ‘𝑇)
168 relfunc 16961 . . . . . . . . . . 11 Rel ((𝑄 ×c 𝑂) Func 𝑇)
169 yoneda.t . . . . . . . . . . . . 13 𝑇 = (SetCat‘𝑉)
170 yoneda.h . . . . . . . . . . . . 13 𝐻 = (HomF𝑄)
171 yoneda.r . . . . . . . . . . . . 13 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
172 yoneda.e . . . . . . . . . . . . 13 𝐸 = (𝑂 evalF 𝑆)
173 yoneda.z . . . . . . . . . . . . 13 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17416, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20yonedalem1 17351 . . . . . . . . . . . 12 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
175174simpld 495 . . . . . . . . . . 11 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
176 1st2ndbr 7597 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
177168, 175, 176sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
17854, 28opelxpd 5481 . . . . . . . . . 10 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
17962, 27opelxpd 5481 . . . . . . . . . 10 (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
180165, 166, 167, 177, 178, 179funcf2 16967 . . . . . . . . 9 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩):(⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)⟶(((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩)))
181164, 43, 10, 14, 118, 54, 28, 62, 27, 166xpchom2 17265 . . . . . . . . . . 11 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
182120xpeq2i 5470 . . . . . . . . . . 11 ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))
183181, 182syl6eq 2847 . . . . . . . . . 10 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋)))
184 df-ov 7019 . . . . . . . . . . . . 13 (𝐹(1st𝑍)𝑋) = ((1st𝑍)‘⟨𝐹, 𝑋⟩)
185 df-ov 7019 . . . . . . . . . . . . 13 (𝐺(1st𝑍)𝑃) = ((1st𝑍)‘⟨𝐺, 𝑃⟩)
186184, 185oveq12i 7028 . . . . . . . . . . . 12 ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)) = (((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩))
187186eqcomi 2804 . . . . . . . . . . 11 (((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃))
188187a1i 11 . . . . . . . . . 10 (𝜑 → (((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)))
189183, 188feq23d 6377 . . . . . . . . 9 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩):(⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)⟶(((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩)) ↔ (⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃))))
190180, 189mpbid 233 . . . . . . . 8 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)))
191190, 34, 30fovrnd 7176 . . . . . . 7 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)))
192 eqid 2795 . . . . . . . . . . 11 (Base‘𝑇) = (Base‘𝑇)
193165, 192, 177funcf1 16965 . . . . . . . . . 10 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
194193, 54, 28fovrnd 7176 . . . . . . . . 9 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ (Base‘𝑇))
195169, 19setcbas 17167 . . . . . . . . 9 (𝜑𝑉 = (Base‘𝑇))
196194, 195eleqtrrd 2886 . . . . . . . 8 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ 𝑉)
197193, 62, 27fovrnd 7176 . . . . . . . . 9 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ (Base‘𝑇))
198197, 195eleqtrrd 2886 . . . . . . . 8 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ 𝑉)
199169, 19, 167, 196, 198elsetchom 17170 . . . . . . 7 (𝜑 → ((𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)) ↔ (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾):(𝐹(1st𝑍)𝑋)⟶(𝐺(1st𝑍)𝑃)))
200191, 199mpbid 233 . . . . . 6 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾):(𝐹(1st𝑍)𝑋)⟶(𝐺(1st𝑍)𝑃))
20116, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28, 62, 27, 34, 30yonedalem22 17357 . . . . . . . 8 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))
2028oppccat 16821 . . . . . . . . . . 11 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
20317, 202syl 17 . . . . . . . . . 10 (𝜑𝑂 ∈ Cat)
20418setccat 17174 . . . . . . . . . . 11 (𝑈 ∈ V → 𝑆 ∈ Cat)
20522, 204syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Cat)
2066, 203, 205fuccat 17069 . . . . . . . . 9 (𝜑𝑄 ∈ Cat)
207170, 206, 43, 14, 45, 54, 82, 62, 12, 31, 34hof2val 17335 . . . . . . . 8 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴) = (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))))
208201, 207eqtrd 2831 . . . . . . 7 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))))
20916, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28yonedalem21 17352 . . . . . . 7 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
21016, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 62, 27yonedalem21 17352 . . . . . . 7 (𝜑 → (𝐺(1st𝑍)𝑃) = (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺))
211208, 209, 210feq123d 6371 . . . . . 6 (𝜑 → ((𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾):(𝐹(1st𝑍)𝑋)⟶(𝐺(1st𝑍)𝑃) ↔ (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶(((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺)))
212200, 211mpbid 233 . . . . 5 (𝜑 → (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶(((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺))
213 eqid 2795 . . . . . 6 (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))) = (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾)))
214213fmpt 6737 . . . . 5 (∀𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾)) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↔ (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶(((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺))
215212, 214sylibr 235 . . . 4 (𝜑 → ∀𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾)) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺))
216 yonedalem3.m . . . . . 6 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
21716, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 62, 27, 216yonedalem3a 17353 . . . . 5 (𝜑 → ((𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))) ∧ (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃)))
218217simpld 495 . . . 4 (𝜑 → (𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))))
219 fveq1 6537 . . . . 5 (𝑎 = ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾)) → (𝑎𝑃) = (((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃))
220219fveq1d 6540 . . . 4 (𝑎 = ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾)) → ((𝑎𝑃)‘( 1𝑃)) = ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃)))
221215, 208, 218, 220fmptcof 6755 . . 3 (𝜑 → ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = (𝑏 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏)(⟨((1st𝑌)‘𝑃), ((1st𝑌)‘𝑋)⟩(comp‘𝑄)𝐺)((𝑃(2nd𝑌)𝑋)‘𝐾))‘𝑃)‘( 1𝑃))))
222 eqid 2795 . . . . . . 7 (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩) = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)
223172, 203, 205, 10, 118, 11, 7, 54, 62, 28, 27, 222, 34, 121evlf2val 17298 . . . . . 6 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) = ((𝐴𝑃)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))((𝑋(2nd𝐹)𝑃)‘𝐾)))
22418, 22, 11, 137, 60, 67, 145, 77setcco 17172 . . . . . 6 (𝜑 → ((𝐴𝑃)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))((𝑋(2nd𝐹)𝑃)‘𝐾)) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
225223, 224eqtrd 2831 . . . . 5 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
226225coeq1d 5618 . . . 4 (𝜑 → ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)) = (((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) ∘ (𝐹𝑀𝑋)))
22716, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28, 216yonedalem3a 17353 . . . . . . . 8 (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋)))
228227simprd 496 . . . . . . 7 (𝜑 → (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋))
229227simpld 495 . . . . . . . 8 (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
230172, 203, 205, 10, 54, 28evlf1 17299 . . . . . . . 8 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
231229, 209, 230feq123d 6371 . . . . . . 7 (𝜑 → ((𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋) ↔ (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st𝐹)‘𝑋)))
232228, 231mpbid 233 . . . . . 6 (𝜑 → (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st𝐹)‘𝑋))
233 eqid 2795 . . . . . . 7 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋)))
234233fmpt 6737 . . . . . 6 (∀𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋) ↔ (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st𝐹)‘𝑋))
235232, 234sylibr 235 . . . . 5 (𝜑 → ∀𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋))
236 fcompt 6758 . . . . . 6 (((𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃) ∧ ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃)) → ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) = (𝑦 ∈ ((1st𝐹)‘𝑋) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦))))
23777, 145, 236syl2anc 584 . . . . 5 (𝜑 → ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) = (𝑦 ∈ ((1st𝐹)‘𝑋) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦))))
238 2fveq3 6543 . . . . 5 (𝑦 = ((𝑎𝑋)‘( 1𝑋)) → ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦)) = ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋)))))
239235, 229, 237, 238fmptcof 6755 . . . 4 (𝜑 → (((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
240226, 239eqtrd 2831 . . 3 (𝜑 → ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
241163, 221, 2403eqtr4d 2841 . 2 (𝜑 → ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)))
242 eqid 2795 . . 3 (comp‘𝑇) = (comp‘𝑇)
243174simprd 496 . . . . . . 7 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
244 1st2ndbr 7597 . . . . . . 7 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
245168, 243, 244sylancr 587 . . . . . 6 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
246165, 192, 245funcf1 16965 . . . . 5 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
247246, 62, 27fovrnd 7176 . . . 4 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ (Base‘𝑇))
248247, 195eleqtrrd 2886 . . 3 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ 𝑉)
249217simprd 496 . . 3 (𝜑 → (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃))
250169, 19, 242, 196, 198, 248, 200, 249setcco 17172 . 2 (𝜑 → ((𝐺𝑀𝑃)(⟨(𝐹(1st𝑍)𝑋), (𝐺(1st𝑍)𝑃)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)))
251246, 54, 28fovrnd 7176 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ (Base‘𝑇))
252251, 195eleqtrrd 2886 . . 3 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ 𝑉)
253165, 166, 167, 245, 178, 179funcf2 16967 . . . . . 6 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩):(⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)⟶(((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)))
254 df-ov 7019 . . . . . . . . . 10 (𝐹(1st𝐸)𝑋) = ((1st𝐸)‘⟨𝐹, 𝑋⟩)
255 df-ov 7019 . . . . . . . . . 10 (𝐺(1st𝐸)𝑃) = ((1st𝐸)‘⟨𝐺, 𝑃⟩)
256254, 255oveq12i 7028 . . . . . . . . 9 ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)) = (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩))
257256eqcomi 2804 . . . . . . . 8 (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃))
258257a1i 11 . . . . . . 7 (𝜑 → (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
259183, 258feq23d 6377 . . . . . 6 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩):(⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)⟶(((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)) ↔ (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃))))
260253, 259mpbid 233 . . . . 5 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
261260, 34, 30fovrnd 7176 . . . 4 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
262169, 19, 167, 252, 248elsetchom 17170 . . . 4 (𝜑 → ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)) ↔ (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾):(𝐹(1st𝐸)𝑋)⟶(𝐺(1st𝐸)𝑃)))
263261, 262mpbid 233 . . 3 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾):(𝐹(1st𝐸)𝑋)⟶(𝐺(1st𝐸)𝑃))
264169, 19, 242, 196, 252, 248, 228, 263setcco 17172 . 2 (𝜑 → ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾)(⟨(𝐹(1st𝑍)𝑋), (𝐹(1st𝐸)𝑋)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐹𝑀𝑋)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)))
265241, 250, 2643eqtr4d 2841 1 (𝜑 → ((𝐺𝑀𝑃)(⟨(𝐹(1st𝑍)𝑋), (𝐺(1st𝑍)𝑃)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾)(⟨(𝐹(1st𝑍)𝑋), (𝐹(1st𝐸)𝑋)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐹𝑀𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  wral 3105  Vcvv 3437  cun 3857  wss 3859  cop 4478   class class class wbr 4962  cmpt 5041   × cxp 5441  ran crn 5444  ccom 5447  Rel wrel 5448  wf 6221  cfv 6225  (class class class)co 7016  cmpo 7018  1st c1st 7543  2nd c2nd 7544  tpos ctpos 7742  Basecbs 16312  Hom chom 16405  compcco 16406  Catccat 16764  Idccid 16765  Homf chomf 16766  oppCatcoppc 16810   Func cfunc 16953  func ccofu 16955   Nat cnat 17040   FuncCat cfuc 17041  SetCatcsetc 17164   ×c cxpc 17247   1stF c1stf 17248   2ndF c2ndf 17249   ⟨,⟩F cprf 17250   evalF cevlf 17288  HomFchof 17327  Yoncyon 17328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-tpos 7743  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-hom 16418  df-cco 16419  df-cat 16768  df-cid 16769  df-homf 16770  df-comf 16771  df-oppc 16811  df-ssc 16909  df-resc 16910  df-subc 16911  df-func 16957  df-cofu 16959  df-nat 17042  df-fuc 17043  df-setc 17165  df-xpc 17251  df-1stf 17252  df-2ndf 17253  df-prf 17254  df-evlf 17292  df-curf 17293  df-hof 17329  df-yon 17330
This theorem is referenced by:  yonedalem3  17359
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