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Theorem yonedalem3b 17120
Description: Lemma for yoneda 17124. (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 6799 . . . . . . . 8 (𝑏 = 𝑎 → (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏) = (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎))
21oveq1d 6806 . . . . . . 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 6332 . . . . . 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 6332 . . . . 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 4884 . . . 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 2771 . . . . . . . . 9 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8 yoneda.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
9 yoneda.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
108, 9oppcbas 16578 . . . . . . . . 9 𝐵 = (Base‘𝑂)
11 eqid 2771 . . . . . . . . 9 (comp‘𝑆) = (comp‘𝑆)
12 eqid 2771 . . . . . . . . 9 (comp‘𝑄) = (comp‘𝑄)
13 eqid 2771 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
146, 7fuchom 16821 . . . . . . . . . . . 12 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
15 relfunc 16722 . . . . . . . . . . . . 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 3942 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
2219, 21ssexd 4939 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ V)
23 yoneda.u . . . . . . . . . . . . . 14 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2416, 17, 8, 18, 6, 22, 23yoncl 17103 . . . . . . . . . . . . 13 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
25 1st2ndbr 7364 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
2615, 24, 25sylancr 575 . . . . . . . . . . . 12 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
27 yonedalem22.p . . . . . . . . . . . 12 (𝜑𝑃𝐵)
28 yonedalem21.x . . . . . . . . . . . 12 (𝜑𝑋𝐵)
299, 13, 14, 26, 27, 28funcf2 16728 . . . . . . . . . . 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 6501 . . . . . . . . . 10 (𝜑 → ((𝑃(2nd𝑌)𝑋)‘𝐾) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st𝑌)‘𝑋)))
3231adantr 466 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑃(2nd𝑌)𝑋)‘𝐾) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st𝑌)‘𝑋)))
33 simpr 471 . . . . . . . . . 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 466 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
366, 7, 12, 33, 35fuccocl 16824 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐺))
3727adantr 466 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑃𝐵)
386, 7, 10, 11, 12, 32, 36, 37fuccoval 16823 . . . . . . . 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 16823 . . . . . . . . . 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 466 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
41 eqid 2771 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
42 relfunc 16722 . . . . . . . . . . . . . . . 16 Rel (𝑂 Func 𝑆)
436fucbas 16820 . . . . . . . . . . . . . . . . . 18 (𝑂 Func 𝑆) = (Base‘𝑄)
449, 43, 26funcf1 16726 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
4544, 28ffvelrnd 6501 . . . . . . . . . . . . . . . 16 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
46 1st2ndbr 7364 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
4742, 45, 46sylancr 575 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
4810, 41, 47funcf1 16726 . . . . . . . . . . . . . 14 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4918, 22setcbas 16928 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝑆))
5049feq3d 6170 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
5148, 50mpbird 247 . . . . . . . . . . . . 13 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
5251, 27ffvelrnd 6501 . . . . . . . . . . . 12 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∈ 𝑈)
5352adantr 466 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∈ 𝑈)
54 yonedalem21.f . . . . . . . . . . . . . . . 16 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
55 1st2ndbr 7364 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
5642, 54, 55sylancr 575 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
5710, 41, 56funcf1 16726 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
5849feq3d 6170 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹):𝐵𝑈 ↔ (1st𝐹):𝐵⟶(Base‘𝑆)))
5957, 58mpbird 247 . . . . . . . . . . . . 13 (𝜑 → (1st𝐹):𝐵𝑈)
6059, 27ffvelrnd 6501 . . . . . . . . . . . 12 (𝜑 → ((1st𝐹)‘𝑃) ∈ 𝑈)
6160adantr 466 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑃) ∈ 𝑈)
62 yonedalem22.g . . . . . . . . . . . . . . . 16 (𝜑𝐺 ∈ (𝑂 Func 𝑆))
63 1st2ndbr 7364 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐺 ∈ (𝑂 Func 𝑆)) → (1st𝐺)(𝑂 Func 𝑆)(2nd𝐺))
6442, 62, 63sylancr 575 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝐺)(𝑂 Func 𝑆)(2nd𝐺))
6510, 41, 64funcf1 16726 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝑆))
6665, 27ffvelrnd 6501 . . . . . . . . . . . . 13 (𝜑 → ((1st𝐺)‘𝑃) ∈ (Base‘𝑆))
6766, 49eleqtrrd 2853 . . . . . . . . . . . 12 (𝜑 → ((1st𝐺)‘𝑃) ∈ 𝑈)
6867adantr 466 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐺)‘𝑃) ∈ 𝑈)
697, 33nat1st2nd 16811 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (⟨(1st ‘((1st𝑌)‘𝑋)), (2nd ‘((1st𝑌)‘𝑋))⟩(𝑂 Nat 𝑆)⟨(1st𝐹), (2nd𝐹)⟩))
70 eqid 2771 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
717, 69, 10, 70, 37natcl 16813 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑃) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑃)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
7218, 40, 70, 53, 61elsetchom 16931 . . . . . . . . . . . 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 222 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃))
747, 34nat1st2nd 16811 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩(𝑂 Nat 𝑆)⟨(1st𝐺), (2nd𝐺)⟩))
757, 74, 10, 70, 27natcl 16813 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑃) ∈ (((1st𝐹)‘𝑃)(Hom ‘𝑆)((1st𝐺)‘𝑃)))
7618, 22, 70, 60, 67elsetchom 16931 . . . . . . . . . . . . 13 (𝜑 → ((𝐴𝑃) ∈ (((1st𝐹)‘𝑃)(Hom ‘𝑆)((1st𝐺)‘𝑃)) ↔ (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃)))
7775, 76mpbid 222 . . . . . . . . . . . 12 (𝜑 → (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃))
7877adantr 466 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃))
7918, 40, 11, 53, 61, 68, 73, 78setcco 16933 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑃), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(𝑎𝑃)) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8039, 79eqtrd 2805 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)‘𝑃) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8180oveq1d 6806 . . . . . . . 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 6501 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝑌)‘𝑃) ∈ (𝑂 Func 𝑆))
83 1st2ndbr 7364 . . . . . . . . . . . . . 14 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑃) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑃)))
8442, 82, 83sylancr 575 . . . . . . . . . . . . 13 (𝜑 → (1st ‘((1st𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑃)))
8510, 41, 84funcf1 16726 . . . . . . . . . . . 12 (𝜑 → (1st ‘((1st𝑌)‘𝑃)):𝐵⟶(Base‘𝑆))
8685, 27ffvelrnd 6501 . . . . . . . . . . 11 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ (Base‘𝑆))
8786, 49eleqtrrd 2853 . . . . . . . . . 10 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
8887adantr 466 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
897, 31nat1st2nd 16811 . . . . . . . . . . . 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 16813 . . . . . . . . . . 11 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃) ∈ (((1st ‘((1st𝑌)‘𝑃))‘𝑃)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
9118, 22, 70, 87, 52elsetchom 16931 . . . . . . . . . . 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 222 . . . . . . . . . 10 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
9392adantr 466 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
94 fco 6196 . . . . . . . . . 10 (((𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃) ∧ (𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃)) → ((𝐴𝑃) ∘ (𝑎𝑃)):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐺)‘𝑃))
9578, 73, 94syl2anc 573 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃) ∘ (𝑎𝑃)):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐺)‘𝑃))
9618, 40, 11, 88, 53, 68, 93, 95setcco 16933 . . . . . . . 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 2809 . . . . . . 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 6332 . . . . . 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 16543 . . . . . . . . 9 (𝜑 → ( 1𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃))
10116, 9, 17, 27, 13, 27yon11 17105 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) = (𝑃(Hom ‘𝐶)𝑃))
102100, 101eleqtrrd 2853 . . . . . . . 8 (𝜑 → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
103102adantr 466 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
104 fvco3 6415 . . . . . . 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 573 . . . . . 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 6501 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑃))
107 fvco3 6415 . . . . . . . 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 573 . . . . . . 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 466 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐶 ∈ Cat)
11028adantr 466 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋𝐵)
111 eqid 2771 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
11230adantr 466 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))
113100adantr 466 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃))
11416, 9, 109, 37, 13, 110, 111, 37, 112, 113yon2 17107 . . . . . . . . . . 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 16545 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐾(⟨𝑃, 𝑃⟩(comp‘𝐶)𝑋)( 1𝑃)) = 𝐾)
116114, 115eqtrd 2805 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) = 𝐾)
117116fveq2d 6334 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = ((𝑎𝑃)‘𝐾))
118 eqid 2771 . . . . . . . . . . . . . . 15 (Hom ‘𝑂) = (Hom ‘𝑂)
11910, 118, 70, 47, 28, 27funcf2 16728 . . . . . . . . . . . . . 14 (𝜑 → (𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12013, 8oppchom 16575 . . . . . . . . . . . . . . 15 (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
12130, 120syl6eleqr 2861 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
122119, 121ffvelrnd 6501 . . . . . . . . . . . . 13 (𝜑 → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12351, 28ffvelrnd 6501 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
12418, 22, 70, 123, 52elsetchom 16931 . . . . . . . . . . . . 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 222 . . . . . . . . . . . 12 (𝜑 → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
126125adantr 466 . . . . . . . . . . 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 16543 . . . . . . . . . . . . 13 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
12816, 9, 17, 28, 13, 28yon11 17105 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
129127, 128eleqtrrd 2853 . . . . . . . . . . . 12 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
130129adantr 466 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
131 fvco3 6415 . . . . . . . . . . 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 573 . . . . . . . . . 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 466 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
1347, 69, 10, 118, 11, 110, 37, 133nati 16815 . . . . . . . . . . . 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 466 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
13618, 40, 11, 135, 53, 61, 126, 73setcco 16933 . . . . . . . . . . . 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 6501 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
138137adantr 466 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
1397, 69, 10, 70, 110natcl 16813 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑋)))
14018, 40, 70, 135, 138elsetchom 16931 . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋))
14210, 118, 70, 56, 28, 27funcf2 16728 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋(2nd𝐹)𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
143142, 121ffvelrnd 6501 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑋(2nd𝐹)𝑃)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
14418, 22, 70, 137, 60elsetchom 16931 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑋(2nd𝐹)𝑃)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)) ↔ ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃)))
145143, 144mpbid 222 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃))
146145adantr 466 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃))
14718, 40, 11, 135, 138, 61, 141, 146setcco 16933 . . . . . . . . . . . 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 2813 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)) = (((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋)))
149148fveq1d 6332 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾))‘( 1𝑋)) = ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)))
150127adantr 466 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
15116, 9, 109, 110, 13, 110, 111, 37, 112, 150yon12 17106 . . . . . . . . . . . 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 16544 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (( 1𝑋)(⟨𝑃, 𝑋⟩(comp‘𝐶)𝑋)𝐾) = 𝐾)
153151, 152eqtrd 2805 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋)) = 𝐾)
154153fveq2d 6334 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘(((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋))) = ((𝑎𝑃)‘𝐾))
155132, 149, 1543eqtr3d 2813 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = ((𝑎𝑃)‘𝐾))
156 fvco3 6415 . . . . . . . . . 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 573 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
158117, 155, 1573eqtr2d 2811 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
159158fveq2d 6334 . . . . . . 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 2805 . . . . . 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 2809 . . . . 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 4878 . . . 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 2817 . . 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 2771 . . . . . . . . . . 11 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
165164, 43, 10xpcbas 17019 . . . . . . . . . 10 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
166 eqid 2771 . . . . . . . . . 10 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
167 eqid 2771 . . . . . . . . . 10 (Hom ‘𝑇) = (Hom ‘𝑇)
168 relfunc 16722 . . . . . . . . . . 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 17113 . . . . . . . . . . . 12 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
175174simpld 482 . . . . . . . . . . 11 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
176 1st2ndbr 7364 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
177168, 175, 176sylancr 575 . . . . . . . . . 10 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
178 opelxpi 5286 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
17954, 28, 178syl2anc 573 . . . . . . . . . 10 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
180 opelxpi 5286 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
18162, 27, 180syl2anc 573 . . . . . . . . . 10 (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
182165, 166, 167, 177, 179, 181funcf2 16728 . . . . . . . . 9 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩):(⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)⟶(((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩)))
183164, 43, 10, 14, 118, 54, 28, 62, 27, 166xpchom2 17027 . . . . . . . . . . 11 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
184120xpeq2i 5275 . . . . . . . . . . 11 ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))
185183, 184syl6eq 2821 . . . . . . . . . 10 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋)))
186 df-ov 6794 . . . . . . . . . . . . 13 (𝐹(1st𝑍)𝑋) = ((1st𝑍)‘⟨𝐹, 𝑋⟩)
187 df-ov 6794 . . . . . . . . . . . . 13 (𝐺(1st𝑍)𝑃) = ((1st𝑍)‘⟨𝐺, 𝑃⟩)
188186, 187oveq12i 6803 . . . . . . . . . . . 12 ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)) = (((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩))
189188eqcomi 2780 . . . . . . . . . . 11 (((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃))
190189a1i 11 . . . . . . . . . 10 (𝜑 → (((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)))
191185, 190feq23d 6178 . . . . . . . . 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))))
192182, 191mpbid 222 . . . . . . . 8 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)))
193192, 34, 30fovrnd 6951 . . . . . . 7 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)))
194 eqid 2771 . . . . . . . . . . 11 (Base‘𝑇) = (Base‘𝑇)
195165, 194, 177funcf1 16726 . . . . . . . . . 10 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
196195, 54, 28fovrnd 6951 . . . . . . . . 9 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ (Base‘𝑇))
197169, 19setcbas 16928 . . . . . . . . 9 (𝜑𝑉 = (Base‘𝑇))
198196, 197eleqtrrd 2853 . . . . . . . 8 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ 𝑉)
199195, 62, 27fovrnd 6951 . . . . . . . . 9 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ (Base‘𝑇))
200199, 197eleqtrrd 2853 . . . . . . . 8 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ 𝑉)
201169, 19, 167, 198, 200elsetchom 16931 . . . . . . 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)))
202193, 201mpbid 222 . . . . . 6 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾):(𝐹(1st𝑍)𝑋)⟶(𝐺(1st𝑍)𝑃))
20316, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28, 62, 27, 34, 30yonedalem22 17119 . . . . . . . 8 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))
2048oppccat 16582 . . . . . . . . . . 11 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
20517, 204syl 17 . . . . . . . . . 10 (𝜑𝑂 ∈ Cat)
20618setccat 16935 . . . . . . . . . . 11 (𝑈 ∈ V → 𝑆 ∈ Cat)
20722, 206syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Cat)
2086, 205, 207fuccat 16830 . . . . . . . . 9 (𝜑𝑄 ∈ Cat)
209170, 208, 43, 14, 45, 54, 82, 62, 12, 31, 34hof2val 17097 . . . . . . . 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))))
210203, 209eqtrd 2805 . . . . . . 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))))
21116, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28yonedalem21 17114 . . . . . . 7 (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
21216, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 62, 27yonedalem21 17114 . . . . . . 7 (𝜑 → (𝐺(1st𝑍)𝑃) = (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺))
213210, 211, 212feq123d 6172 . . . . . 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)))
214202, 213mpbid 222 . . . . 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))
215 eqid 2771 . . . . . 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)))
216215fmpt 6521 . . . . 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))
217214, 216sylibr 224 . . . 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))
218 yonedalem3.m . . . . . 6 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
21916, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 62, 27, 218yonedalem3a 17115 . . . . 5 (𝜑 → ((𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))) ∧ (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃)))
220219simpld 482 . . . 4 (𝜑 → (𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))))
221 fveq1 6329 . . . . 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))
222221fveq1d 6332 . . . 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)))
223217, 210, 220, 222fmptcof 6538 . . 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))))
224 eqid 2771 . . . . . . 7 (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩) = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)
225172, 205, 207, 10, 118, 11, 7, 54, 62, 28, 27, 224, 34, 121evlf2val 17060 . . . . . 6 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) = ((𝐴𝑃)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))((𝑋(2nd𝐹)𝑃)‘𝐾)))
22618, 22, 11, 137, 60, 67, 145, 77setcco 16933 . . . . . 6 (𝜑 → ((𝐴𝑃)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))((𝑋(2nd𝐹)𝑃)‘𝐾)) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
227225, 226eqtrd 2805 . . . . 5 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
228227coeq1d 5420 . . . 4 (𝜑 → ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)) = (((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) ∘ (𝐹𝑀𝑋)))
22916, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28, 218yonedalem3a 17115 . . . . . . . 8 (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋)))
230229simprd 483 . . . . . . 7 (𝜑 → (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋))
231229simpld 482 . . . . . . . 8 (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
232172, 205, 207, 10, 54, 28evlf1 17061 . . . . . . . 8 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
233231, 211, 232feq123d 6172 . . . . . . 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)))
234230, 233mpbid 222 . . . . . 6 (𝜑 → (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st𝐹)‘𝑋))
235 eqid 2771 . . . . . . 7 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋)))
236235fmpt 6521 . . . . . 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))
237234, 236sylibr 224 . . . . 5 (𝜑 → ∀𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋))
238 fcompt 6541 . . . . . 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))))
23977, 145, 238syl2anc 573 . . . . 5 (𝜑 → ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) = (𝑦 ∈ ((1st𝐹)‘𝑋) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦))))
240 fveq2 6330 . . . . . 6 (𝑦 = ((𝑎𝑋)‘( 1𝑋)) → (((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
241240fveq2d 6334 . . . . 5 (𝑦 = ((𝑎𝑋)‘( 1𝑋)) → ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦)) = ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋)))))
242237, 231, 239, 241fmptcof 6538 . . . 4 (𝜑 → (((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
243228, 242eqtrd 2805 . . 3 (𝜑 → ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
244163, 223, 2433eqtr4d 2815 . 2 (𝜑 → ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)))
245 eqid 2771 . . 3 (comp‘𝑇) = (comp‘𝑇)
246174simprd 483 . . . . . . 7 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
247 1st2ndbr 7364 . . . . . . 7 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
248168, 246, 247sylancr 575 . . . . . 6 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
249165, 194, 248funcf1 16726 . . . . 5 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
250249, 62, 27fovrnd 6951 . . . 4 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ (Base‘𝑇))
251250, 197eleqtrrd 2853 . . 3 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ 𝑉)
252219simprd 483 . . 3 (𝜑 → (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃))
253169, 19, 245, 198, 200, 251, 202, 252setcco 16933 . 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)))
254249, 54, 28fovrnd 6951 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ (Base‘𝑇))
255254, 197eleqtrrd 2853 . . 3 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ 𝑉)
256165, 166, 167, 248, 179, 181funcf2 16728 . . . . . 6 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩):(⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)⟶(((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)))
257 df-ov 6794 . . . . . . . . . 10 (𝐹(1st𝐸)𝑋) = ((1st𝐸)‘⟨𝐹, 𝑋⟩)
258 df-ov 6794 . . . . . . . . . 10 (𝐺(1st𝐸)𝑃) = ((1st𝐸)‘⟨𝐺, 𝑃⟩)
259257, 258oveq12i 6803 . . . . . . . . 9 ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)) = (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩))
260259eqcomi 2780 . . . . . . . 8 (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃))
261260a1i 11 . . . . . . 7 (𝜑 → (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
262185, 261feq23d 6178 . . . . . 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))))
263256, 262mpbid 222 . . . . 5 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
264263, 34, 30fovrnd 6951 . . . 4 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
265169, 19, 167, 255, 251elsetchom 16931 . . . 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)))
266264, 265mpbid 222 . . 3 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾):(𝐹(1st𝐸)𝑋)⟶(𝐺(1st𝐸)𝑃))
267169, 19, 245, 198, 255, 251, 230, 266setcco 16933 . 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)))
268244, 253, 2673eqtr4d 2815 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 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cun 3721  wss 3723  cop 4322   class class class wbr 4786  cmpt 4863   × cxp 5247  ran crn 5250  ccom 5253  Rel wrel 5254  wf 6025  cfv 6029  (class class class)co 6791  cmpt2 6793  1st c1st 7311  2nd c2nd 7312  tpos ctpos 7501  Basecbs 16057  Hom chom 16153  compcco 16154  Catccat 16525  Idccid 16526  Homf chomf 16527  oppCatcoppc 16571   Func cfunc 16714  func ccofu 16716   Nat cnat 16801   FuncCat cfuc 16802  SetCatcsetc 16925   ×c cxpc 17009   1stF c1stf 17010   2ndF c2ndf 17011   ⟨,⟩F cprf 17012   evalF cevlf 17050  HomFchof 17089  Yoncyon 17090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-tpos 7502  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-pm 8010  df-ixp 8061  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-2 11279  df-3 11280  df-4 11281  df-5 11282  df-6 11283  df-7 11284  df-8 11285  df-9 11286  df-n0 11493  df-z 11578  df-dec 11694  df-uz 11887  df-fz 12527  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-hom 16167  df-cco 16168  df-cat 16529  df-cid 16530  df-homf 16531  df-comf 16532  df-oppc 16572  df-ssc 16670  df-resc 16671  df-subc 16672  df-func 16718  df-cofu 16720  df-nat 16803  df-fuc 16804  df-setc 16926  df-xpc 17013  df-1stf 17014  df-2ndf 17015  df-prf 17016  df-evlf 17054  df-curf 17055  df-hof 17091  df-yon 17092
This theorem is referenced by:  yonedalem3  17121
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