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Theorem yonedalem3b 18335
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))
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 7438 . . . . . . . 8 (𝑏 = 𝑎 → (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏) = (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎))
21oveq1d 7445 . . . . . . 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 6908 . . . . . 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 6908 . . . . 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 5260 . . . 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 2734 . . . . . . . . 9 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8 yoneda.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
9 yoneda.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
108, 9oppcbas 17763 . . . . . . . . 9 𝐵 = (Base‘𝑂)
11 eqid 2734 . . . . . . . . 9 (comp‘𝑆) = (comp‘𝑆)
12 eqid 2734 . . . . . . . . 9 (comp‘𝑄) = (comp‘𝑄)
13 eqid 2734 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
146, 7fuchom 18016 . . . . . . . . . . . 12 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
15 relfunc 17912 . . . . . . . . . . . . 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 4203 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
2219, 21ssexd 5329 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ V)
23 yoneda.u . . . . . . . . . . . . . 14 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2416, 17, 8, 18, 6, 22, 23yoncl 18318 . . . . . . . . . . . . 13 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
25 1st2ndbr 8065 . . . . . . . . . . . . 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 17918 . . . . . . . . . . 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, 30ffvelcdmd 7104 . . . . . . . . . 10 (𝜑 → ((𝑃(2nd𝑌)𝑋)‘𝐾) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st𝑌)‘𝑋)))
3231adantr 480 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑃(2nd𝑌)𝑋)‘𝐾) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st𝑌)‘𝑋)))
33 simpr 484 . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
366, 7, 12, 33, 35fuccocl 18020 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐺))
3727adantr 480 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑃𝐵)
386, 7, 10, 11, 12, 32, 36, 37fuccoval 18019 . . . . . . . 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 18019 . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
41 eqid 2734 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
42 relfunc 17912 . . . . . . . . . . . . . . . 16 Rel (𝑂 Func 𝑆)
436fucbas 18015 . . . . . . . . . . . . . . . . . 18 (𝑂 Func 𝑆) = (Base‘𝑄)
449, 43, 26funcf1 17916 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
4544, 28ffvelcdmd 7104 . . . . . . . . . . . . . . . 16 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
46 1st2ndbr 8065 . . . . . . . . . . . . . . . 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 17916 . . . . . . . . . . . . . 14 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4918, 22setcbas 18131 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝑆))
5049feq3d 6723 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
5148, 50mpbird 257 . . . . . . . . . . . . 13 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
5251, 27ffvelcdmd 7104 . . . . . . . . . . . 12 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∈ 𝑈)
5352adantr 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∈ 𝑈)
54 yonedalem21.f . . . . . . . . . . . . . . . 16 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
55 1st2ndbr 8065 . . . . . . . . . . . . . . . 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 17916 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
5849feq3d 6723 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹):𝐵𝑈 ↔ (1st𝐹):𝐵⟶(Base‘𝑆)))
5957, 58mpbird 257 . . . . . . . . . . . . 13 (𝜑 → (1st𝐹):𝐵𝑈)
6059, 27ffvelcdmd 7104 . . . . . . . . . . . 12 (𝜑 → ((1st𝐹)‘𝑃) ∈ 𝑈)
6160adantr 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑃) ∈ 𝑈)
62 yonedalem22.g . . . . . . . . . . . . . . . 16 (𝜑𝐺 ∈ (𝑂 Func 𝑆))
63 1st2ndbr 8065 . . . . . . . . . . . . . . . 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 17916 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝑆))
6665, 27ffvelcdmd 7104 . . . . . . . . . . . . 13 (𝜑 → ((1st𝐺)‘𝑃) ∈ (Base‘𝑆))
6766, 49eleqtrrd 2841 . . . . . . . . . . . 12 (𝜑 → ((1st𝐺)‘𝑃) ∈ 𝑈)
6867adantr 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐺)‘𝑃) ∈ 𝑈)
697, 33nat1st2nd 18005 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (⟨(1st ‘((1st𝑌)‘𝑋)), (2nd ‘((1st𝑌)‘𝑋))⟩(𝑂 Nat 𝑆)⟨(1st𝐹), (2nd𝐹)⟩))
70 eqid 2734 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
717, 69, 10, 70, 37natcl 18007 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑃) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑃)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
7218, 40, 70, 53, 61elsetchom 18134 . . . . . . . . . . . 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 232 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃))
747, 34nat1st2nd 18005 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩(𝑂 Nat 𝑆)⟨(1st𝐺), (2nd𝐺)⟩))
757, 74, 10, 70, 27natcl 18007 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑃) ∈ (((1st𝐹)‘𝑃)(Hom ‘𝑆)((1st𝐺)‘𝑃)))
7618, 22, 70, 60, 67elsetchom 18134 . . . . . . . . . . . . 13 (𝜑 → ((𝐴𝑃) ∈ (((1st𝐹)‘𝑃)(Hom ‘𝑆)((1st𝐺)‘𝑃)) ↔ (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃)))
7775, 76mpbid 232 . . . . . . . . . . . 12 (𝜑 → (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃))
7877adantr 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃))
7918, 40, 11, 53, 61, 68, 73, 78setcco 18136 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑃), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(𝑎𝑃)) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8039, 79eqtrd 2774 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)‘𝑃) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8180oveq1d 7445 . . . . . . . 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, 27ffvelcdmd 7104 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝑌)‘𝑃) ∈ (𝑂 Func 𝑆))
83 1st2ndbr 8065 . . . . . . . . . . . . . 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 17916 . . . . . . . . . . . 12 (𝜑 → (1st ‘((1st𝑌)‘𝑃)):𝐵⟶(Base‘𝑆))
8685, 27ffvelcdmd 7104 . . . . . . . . . . 11 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ (Base‘𝑆))
8786, 49eleqtrrd 2841 . . . . . . . . . 10 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
8887adantr 480 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
897, 31nat1st2nd 18005 . . . . . . . . . . . 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 18007 . . . . . . . . . . 11 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃) ∈ (((1st ‘((1st𝑌)‘𝑃))‘𝑃)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
9118, 22, 70, 87, 52elsetchom 18134 . . . . . . . . . . 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 232 . . . . . . . . . 10 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
9392adantr 480 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
94 fco 6760 . . . . . . . . . 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 18136 . . . . . . . 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 2778 . . . . . . 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 6908 . . . . . 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 17726 . . . . . . . . 9 (𝜑 → ( 1𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃))
10116, 9, 17, 27, 13, 27yon11 18320 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) = (𝑃(Hom ‘𝐶)𝑃))
102100, 101eleqtrrd 2841 . . . . . . . 8 (𝜑 → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
103102adantr 480 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
104 fvco3 7007 . . . . . . 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, 103ffvelcdmd 7104 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑃))
107 fvco3 7007 . . . . . . . 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 480 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐶 ∈ Cat)
11028adantr 480 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋𝐵)
111 eqid 2734 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
11230adantr 480 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))
113100adantr 480 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃))
11416, 9, 109, 37, 13, 110, 111, 37, 112, 113yon2 18322 . . . . . . . . . . 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 17728 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐾(⟨𝑃, 𝑃⟩(comp‘𝐶)𝑋)( 1𝑃)) = 𝐾)
116114, 115eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) = 𝐾)
117116fveq2d 6910 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = ((𝑎𝑃)‘𝐾))
118 eqid 2734 . . . . . . . . . . . . . . 15 (Hom ‘𝑂) = (Hom ‘𝑂)
11910, 118, 70, 47, 28, 27funcf2 17918 . . . . . . . . . . . . . 14 (𝜑 → (𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12013, 8oppchom 17760 . . . . . . . . . . . . . . 15 (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
12130, 120eleqtrrdi 2849 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
122119, 121ffvelcdmd 7104 . . . . . . . . . . . . 13 (𝜑 → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12351, 28ffvelcdmd 7104 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
12418, 22, 70, 123, 52elsetchom 18134 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 12 (𝜑 → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
126125adantr 480 . . . . . . . . . . 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 17726 . . . . . . . . . . . . 13 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
12816, 9, 17, 28, 13, 28yon11 18320 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
129127, 128eleqtrrd 2841 . . . . . . . . . . . 12 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
130129adantr 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
131 fvco3 7007 . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
1347, 69, 10, 118, 11, 110, 37, 133nati 18009 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
13618, 40, 11, 135, 53, 61, 126, 73setcco 18136 . . . . . . . . . . . 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, 28ffvelcdmd 7104 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
138137adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
1397, 69, 10, 70, 110natcl 18007 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑋)))
14018, 40, 70, 135, 138elsetchom 18134 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋))
14210, 118, 70, 56, 28, 27funcf2 17918 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋(2nd𝐹)𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
143142, 121ffvelcdmd 7104 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑋(2nd𝐹)𝑃)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
14418, 22, 70, 137, 60elsetchom 18134 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑋(2nd𝐹)𝑃)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)) ↔ ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃)))
145143, 144mpbid 232 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃))
146145adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃))
14718, 40, 11, 135, 138, 61, 141, 146setcco 18136 . . . . . . . . . . . 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 2782 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)) = (((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋)))
149148fveq1d 6908 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾))‘( 1𝑋)) = ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)))
150127adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
15116, 9, 109, 110, 13, 110, 111, 37, 112, 150yon12 18321 . . . . . . . . . . . 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 17727 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (( 1𝑋)(⟨𝑃, 𝑋⟩(comp‘𝐶)𝑋)𝐾) = 𝐾)
153151, 152eqtrd 2774 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋)) = 𝐾)
154153fveq2d 6910 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘(((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋))) = ((𝑎𝑃)‘𝐾))
155132, 149, 1543eqtr3d 2782 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = ((𝑎𝑃)‘𝐾))
156 fvco3 7007 . . . . . . . . . 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 2780 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
159158fveq2d 6910 . . . . . . 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 2774 . . . . . 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 2778 . . . . 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 5247 . . . 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, 162eqtrid 2786 . . 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 2734 . . . . . . . . . . 11 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
165164, 43, 10xpcbas 18233 . . . . . . . . . 10 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
166 eqid 2734 . . . . . . . . . 10 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
167 eqid 2734 . . . . . . . . . 10 (Hom ‘𝑇) = (Hom ‘𝑇)
168 relfunc 17912 . . . . . . . . . . 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 18328 . . . . . . . . . . . 12 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
175174simpld 494 . . . . . . . . . . 11 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
176 1st2ndbr 8065 . . . . . . . . . . 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 5727 . . . . . . . . . 10 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
17962, 27opelxpd 5727 . . . . . . . . . 10 (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
180165, 166, 167, 177, 178, 179funcf2 17918 . . . . . . . . 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 18241 . . . . . . . . . . 11 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
182120xpeq2i 5715 . . . . . . . . . . 11 ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))
183181, 182eqtrdi 2790 . . . . . . . . . 10 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋)))
184 df-ov 7433 . . . . . . . . . . . . 13 (𝐹(1st𝑍)𝑋) = ((1st𝑍)‘⟨𝐹, 𝑋⟩)
185 df-ov 7433 . . . . . . . . . . . . 13 (𝐺(1st𝑍)𝑃) = ((1st𝑍)‘⟨𝐺, 𝑃⟩)
186184, 185oveq12i 7442 . . . . . . . . . . . 12 ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)) = (((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩))
187186eqcomi 2743 . . . . . . . . . . 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 6731 . . . . . . . . 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 232 . . . . . . . 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, 30fovcdmd 7604 . . . . . . 7 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)))
192 eqid 2734 . . . . . . . . . . 11 (Base‘𝑇) = (Base‘𝑇)
193165, 192, 177funcf1 17916 . . . . . . . . . 10 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
194193, 54, 28fovcdmd 7604 . . . . . . . . 9 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ (Base‘𝑇))
195169, 19setcbas 18131 . . . . . . . . 9 (𝜑𝑉 = (Base‘𝑇))
196194, 195eleqtrrd 2841 . . . . . . . 8 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ 𝑉)
197193, 62, 27fovcdmd 7604 . . . . . . . . 9 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ (Base‘𝑇))
198197, 195eleqtrrd 2841 . . . . . . . 8 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ 𝑉)
199169, 19, 167, 196, 198elsetchom 18134 . . . . . . 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 232 . . . . . 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 18334 . . . . . . . 8 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))
2028oppccat 17768 . . . . . . . . . . 11 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
20317, 202syl 17 . . . . . . . . . 10 (𝜑𝑂 ∈ Cat)
20418setccat 18138 . . . . . . . . . . 11 (𝑈 ∈ V → 𝑆 ∈ Cat)
20522, 204syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Cat)
2066, 203, 205fuccat 18026 . . . . . . . . 9 (𝜑𝑄 ∈ Cat)
207170, 206, 43, 14, 45, 54, 82, 62, 12, 31, 34hof2val 18312 . . . . . . . 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 2774 . . . . . . 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 18329 . . . . . . 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 18329 . . . . . . 7 (𝜑 → (𝐺(1st𝑍)𝑃) = (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺))
211208, 209, 210feq123d 6725 . . . . . 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 232 . . . . 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 2734 . . . . . 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 7129 . . . . 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 234 . . . 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 18330 . . . . 5 (𝜑 → ((𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))) ∧ (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃)))
218217simpld 494 . . . 4 (𝜑 → (𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))))
219 fveq1 6905 . . . . 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 6908 . . . 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 7149 . . 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 2734 . . . . . . 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 18275 . . . . . 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 18136 . . . . . 6 (𝜑 → ((𝐴𝑃)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))((𝑋(2nd𝐹)𝑃)‘𝐾)) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
225223, 224eqtrd 2774 . . . . 5 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
226225coeq1d 5874 . . . 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 18330 . . . . . . . 8 (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋)))
228227simprd 495 . . . . . . 7 (𝜑 → (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋))
229227simpld 494 . . . . . . . 8 (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
230172, 203, 205, 10, 54, 28evlf1 18276 . . . . . . . 8 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
231229, 209, 230feq123d 6725 . . . . . . 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 232 . . . . . 6 (𝜑 → (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st𝐹)‘𝑋))
233 eqid 2734 . . . . . . 7 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋)))
234233fmpt 7129 . . . . . 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 234 . . . . 5 (𝜑 → ∀𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋))
236 fcompt 7152 . . . . . 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 6911 . . . . 5 (𝑦 = ((𝑎𝑋)‘( 1𝑋)) → ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦)) = ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋)))))
239235, 229, 237, 238fmptcof 7149 . . . 4 (𝜑 → (((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
240226, 239eqtrd 2774 . . 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 2784 . 2 (𝜑 → ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)))
242 eqid 2734 . . 3 (comp‘𝑇) = (comp‘𝑇)
243174simprd 495 . . . . . . 7 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
244 1st2ndbr 8065 . . . . . . 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 17916 . . . . 5 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
247246, 62, 27fovcdmd 7604 . . . 4 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ (Base‘𝑇))
248247, 195eleqtrrd 2841 . . 3 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ 𝑉)
249217simprd 495 . . 3 (𝜑 → (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃))
250169, 19, 242, 196, 198, 248, 200, 249setcco 18136 . 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, 28fovcdmd 7604 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ (Base‘𝑇))
252251, 195eleqtrrd 2841 . . 3 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ 𝑉)
253165, 166, 167, 245, 178, 179funcf2 17918 . . . . . 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 7433 . . . . . . . . . 10 (𝐹(1st𝐸)𝑋) = ((1st𝐸)‘⟨𝐹, 𝑋⟩)
255 df-ov 7433 . . . . . . . . . 10 (𝐺(1st𝐸)𝑃) = ((1st𝐸)‘⟨𝐺, 𝑃⟩)
256254, 255oveq12i 7442 . . . . . . . . 9 ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)) = (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩))
257256eqcomi 2743 . . . . . . . 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 6731 . . . . . 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 232 . . . . 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, 30fovcdmd 7604 . . . 4 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
262169, 19, 167, 252, 248elsetchom 18134 . . . 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 232 . . 3 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾):(𝐹(1st𝐸)𝑋)⟶(𝐺(1st𝐸)𝑃))
264169, 19, 242, 196, 252, 248, 228, 263setcco 18136 . 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 2784 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 395   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  cun 3960  wss 3962  cop 4636   class class class wbr 5147  cmpt 5230   × cxp 5686  ran crn 5689  ccom 5692  Rel wrel 5693  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  1st c1st 8010  2nd c2nd 8011  tpos ctpos 8248  Basecbs 17244  Hom chom 17308  compcco 17309  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-pm 8867  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-ress 17274  df-hom 17321  df-cco 17322  df-cat 17712  df-cid 17713  df-homf 17714  df-comf 17715  df-oppc 17756  df-ssc 17857  df-resc 17858  df-subc 17859  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-evlf 18269  df-curf 18270  df-hof 18306  df-yon 18307
This theorem is referenced by:  yonedalem3  18336
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