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Theorem yonedalem3b 16966
Description: Lemma for yoneda 16970. (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 6698 . . . . . . . 8 (𝑏 = 𝑎 → (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏) = (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎))
21oveq1d 6705 . . . . . . 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 6231 . . . . . 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 6231 . . . . 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 4783 . . . 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 2651 . . . . . . . . 9 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8 yoneda.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
9 yoneda.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
108, 9oppcbas 16425 . . . . . . . . 9 𝐵 = (Base‘𝑂)
11 eqid 2651 . . . . . . . . 9 (comp‘𝑆) = (comp‘𝑆)
12 eqid 2651 . . . . . . . . 9 (comp‘𝑄) = (comp‘𝑄)
13 eqid 2651 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
146, 7fuchom 16668 . . . . . . . . . . . 12 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
15 relfunc 16569 . . . . . . . . . . . . 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 3824 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
2219, 21ssexd 4838 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ V)
23 yoneda.u . . . . . . . . . . . . . 14 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2416, 17, 8, 18, 6, 22, 23yoncl 16949 . . . . . . . . . . . . 13 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
25 1st2ndbr 7261 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
2615, 24, 25sylancr 696 . . . . . . . . . . . 12 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
27 yonedalem22.p . . . . . . . . . . . 12 (𝜑𝑃𝐵)
28 yonedalem21.x . . . . . . . . . . . 12 (𝜑𝑋𝐵)
299, 13, 14, 26, 27, 28funcf2 16575 . . . . . . . . . . 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 6400 . . . . . . . . . 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 476 . . . . . . . . . 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 16671 . . . . . . . . 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 16670 . . . . . . . 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 16670 . . . . . . . . . 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 2651 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
42 relfunc 16569 . . . . . . . . . . . . . . . 16 Rel (𝑂 Func 𝑆)
436fucbas 16667 . . . . . . . . . . . . . . . . . 18 (𝑂 Func 𝑆) = (Base‘𝑄)
449, 43, 26funcf1 16573 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
4544, 28ffvelrnd 6400 . . . . . . . . . . . . . . . 16 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
46 1st2ndbr 7261 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
4742, 45, 46sylancr 696 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
4810, 41, 47funcf1 16573 . . . . . . . . . . . . . 14 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4918, 22setcbas 16775 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝑆))
5049feq3d 6070 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
5148, 50mpbird 247 . . . . . . . . . . . . 13 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
5251, 27ffvelrnd 6400 . . . . . . . . . . . 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 7261 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
5642, 54, 55sylancr 696 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
5710, 41, 56funcf1 16573 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
5849feq3d 6070 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹):𝐵𝑈 ↔ (1st𝐹):𝐵⟶(Base‘𝑆)))
5957, 58mpbird 247 . . . . . . . . . . . . 13 (𝜑 → (1st𝐹):𝐵𝑈)
6059, 27ffvelrnd 6400 . . . . . . . . . . . 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 7261 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐺 ∈ (𝑂 Func 𝑆)) → (1st𝐺)(𝑂 Func 𝑆)(2nd𝐺))
6442, 62, 63sylancr 696 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝐺)(𝑂 Func 𝑆)(2nd𝐺))
6510, 41, 64funcf1 16573 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝑆))
6665, 27ffvelrnd 6400 . . . . . . . . . . . . 13 (𝜑 → ((1st𝐺)‘𝑃) ∈ (Base‘𝑆))
6766, 49eleqtrrd 2733 . . . . . . . . . . . 12 (𝜑 → ((1st𝐺)‘𝑃) ∈ 𝑈)
6867adantr 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐺)‘𝑃) ∈ 𝑈)
697, 33nat1st2nd 16658 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (⟨(1st ‘((1st𝑌)‘𝑋)), (2nd ‘((1st𝑌)‘𝑋))⟩(𝑂 Nat 𝑆)⟨(1st𝐹), (2nd𝐹)⟩))
70 eqid 2651 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
717, 69, 10, 70, 37natcl 16660 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑃) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑃)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
7218, 40, 70, 53, 61elsetchom 16778 . . . . . . . . . . . 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 16658 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩(𝑂 Nat 𝑆)⟨(1st𝐺), (2nd𝐺)⟩))
757, 74, 10, 70, 27natcl 16660 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑃) ∈ (((1st𝐹)‘𝑃)(Hom ‘𝑆)((1st𝐺)‘𝑃)))
7618, 22, 70, 60, 67elsetchom 16778 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃))
7918, 40, 11, 53, 61, 68, 73, 78setcco 16780 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑃), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(𝑎𝑃)) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8039, 79eqtrd 2685 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)‘𝑃) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8180oveq1d 6705 . . . . . . . 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 6400 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝑌)‘𝑃) ∈ (𝑂 Func 𝑆))
83 1st2ndbr 7261 . . . . . . . . . . . . . 14 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑃) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑃)))
8442, 82, 83sylancr 696 . . . . . . . . . . . . 13 (𝜑 → (1st ‘((1st𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑃)))
8510, 41, 84funcf1 16573 . . . . . . . . . . . 12 (𝜑 → (1st ‘((1st𝑌)‘𝑃)):𝐵⟶(Base‘𝑆))
8685, 27ffvelrnd 6400 . . . . . . . . . . 11 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ (Base‘𝑆))
8786, 49eleqtrrd 2733 . . . . . . . . . 10 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
8887adantr 480 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
897, 31nat1st2nd 16658 . . . . . . . . . . . 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 16660 . . . . . . . . . . 11 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃) ∈ (((1st ‘((1st𝑌)‘𝑃))‘𝑃)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
9118, 22, 70, 87, 52elsetchom 16778 . . . . . . . . . . 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 480 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
94 fco 6096 . . . . . . . . . 10 (((𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃) ∧ (𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃)) → ((𝐴𝑃) ∘ (𝑎𝑃)):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐺)‘𝑃))
9578, 73, 94syl2anc 694 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃) ∘ (𝑎𝑃)):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐺)‘𝑃))
9618, 40, 11, 88, 53, 68, 93, 95setcco 16780 . . . . . . . 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 2689 . . . . . . 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 6231 . . . . . 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 16390 . . . . . . . . 9 (𝜑 → ( 1𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃))
10116, 9, 17, 27, 13, 27yon11 16951 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) = (𝑃(Hom ‘𝐶)𝑃))
102100, 101eleqtrrd 2733 . . . . . . . 8 (𝜑 → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
103102adantr 480 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
104 fvco3 6314 . . . . . . 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 694 . . . . . 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 6400 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑃))
107 fvco3 6314 . . . . . . . 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 694 . . . . . . 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 2651 . . . . . . . . . . . 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 16953 . . . . . . . . . . 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 16392 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐾(⟨𝑃, 𝑃⟩(comp‘𝐶)𝑋)( 1𝑃)) = 𝐾)
116114, 115eqtrd 2685 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) = 𝐾)
117116fveq2d 6233 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = ((𝑎𝑃)‘𝐾))
118 eqid 2651 . . . . . . . . . . . . . . 15 (Hom ‘𝑂) = (Hom ‘𝑂)
11910, 118, 70, 47, 28, 27funcf2 16575 . . . . . . . . . . . . . 14 (𝜑 → (𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12013, 8oppchom 16422 . . . . . . . . . . . . . . 15 (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
12130, 120syl6eleqr 2741 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
122119, 121ffvelrnd 6400 . . . . . . . . . . . . 13 (𝜑 → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12351, 28ffvelrnd 6400 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
12418, 22, 70, 123, 52elsetchom 16778 . . . . . . . . . . . . 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 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 16390 . . . . . . . . . . . . 13 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
12816, 9, 17, 28, 13, 28yon11 16951 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
129127, 128eleqtrrd 2733 . . . . . . . . . . . 12 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
130129adantr 480 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
131 fvco3 6314 . . . . . . . . . . 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 694 . . . . . . . . . 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 16662 . . . . . . . . . . . 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 16780 . . . . . . . . . . . 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 6400 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
138137adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
1397, 69, 10, 70, 110natcl 16660 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑋)))
14018, 40, 70, 135, 138elsetchom 16778 . . . . . . . . . . . . . 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 16575 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋(2nd𝐹)𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
143142, 121ffvelrnd 6400 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑋(2nd𝐹)𝑃)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
14418, 22, 70, 137, 60elsetchom 16778 . . . . . . . . . . . . . . 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 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 16780 . . . . . . . . . . . 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 2693 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)) = (((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋)))
149148fveq1d 6231 . . . . . . . . . 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 16952 . . . . . . . . . . . 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 16391 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (( 1𝑋)(⟨𝑃, 𝑋⟩(comp‘𝐶)𝑋)𝐾) = 𝐾)
153151, 152eqtrd 2685 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋)) = 𝐾)
154153fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘(((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋))) = ((𝑎𝑃)‘𝐾))
155132, 149, 1543eqtr3d 2693 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = ((𝑎𝑃)‘𝐾))
156 fvco3 6314 . . . . . . . . . 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 694 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
158117, 155, 1573eqtr2d 2691 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
159158fveq2d 6233 . . . . . . 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 2685 . . . . . 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 2689 . . . . 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 4777 . . . 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 2697 . . 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 2651 . . . . . . . . . . 11 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
165164, 43, 10xpcbas 16865 . . . . . . . . . 10 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
166 eqid 2651 . . . . . . . . . 10 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
167 eqid 2651 . . . . . . . . . 10 (Hom ‘𝑇) = (Hom ‘𝑇)
168 relfunc 16569 . . . . . . . . . . 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 16959 . . . . . . . . . . . 12 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
175174simpld 474 . . . . . . . . . . 11 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
176 1st2ndbr 7261 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
177168, 175, 176sylancr 696 . . . . . . . . . 10 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
178 opelxpi 5182 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
17954, 28, 178syl2anc 694 . . . . . . . . . 10 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
180 opelxpi 5182 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
18162, 27, 180syl2anc 694 . . . . . . . . . 10 (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
182165, 166, 167, 177, 179, 181funcf2 16575 . . . . . . . . 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 16873 . . . . . . . . . . 11 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
184120xpeq2i 5170 . . . . . . . . . . 11 ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))
185183, 184syl6eq 2701 . . . . . . . . . 10 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋)))
186 df-ov 6693 . . . . . . . . . . . . 13 (𝐹(1st𝑍)𝑋) = ((1st𝑍)‘⟨𝐹, 𝑋⟩)
187 df-ov 6693 . . . . . . . . . . . . 13 (𝐺(1st𝑍)𝑃) = ((1st𝑍)‘⟨𝐺, 𝑃⟩)
188186, 187oveq12i 6702 . . . . . . . . . . . 12 ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)) = (((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩))
189188eqcomi 2660 . . . . . . . . . . 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 6078 . . . . . . . . 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 6848 . . . . . . 7 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)))
194 eqid 2651 . . . . . . . . . . 11 (Base‘𝑇) = (Base‘𝑇)
195165, 194, 177funcf1 16573 . . . . . . . . . 10 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
196195, 54, 28fovrnd 6848 . . . . . . . . 9 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ (Base‘𝑇))
197169, 19setcbas 16775 . . . . . . . . 9 (𝜑𝑉 = (Base‘𝑇))
198196, 197eleqtrrd 2733 . . . . . . . 8 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ 𝑉)
199195, 62, 27fovrnd 6848 . . . . . . . . 9 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ (Base‘𝑇))
200199, 197eleqtrrd 2733 . . . . . . . 8 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ 𝑉)
201169, 19, 167, 198, 200elsetchom 16778 . . . . . . 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 16965 . . . . . . . 8 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))
2048oppccat 16429 . . . . . . . . . . 11 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
20517, 204syl 17 . . . . . . . . . 10 (𝜑𝑂 ∈ Cat)
20618setccat 16782 . . . . . . . . . . 11 (𝑈 ∈ V → 𝑆 ∈ Cat)
20722, 206syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Cat)
2086, 205, 207fuccat 16677 . . . . . . . . 9 (𝜑𝑄 ∈ Cat)
209170, 208, 43, 14, 45, 54, 82, 62, 12, 31, 34hof2val 16943 . . . . . . . 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 2685 . . . . . . 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 16960 . . . . . . 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 16960 . . . . . . 7 (𝜑 → (𝐺(1st𝑍)𝑃) = (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺))
213210, 211, 212feq123d 6072 . . . . . 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 2651 . . . . . 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 6421 . . . . 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 16961 . . . . 5 (𝜑 → ((𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))) ∧ (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃)))
220219simpld 474 . . . 4 (𝜑 → (𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))))
221 fveq1 6228 . . . . 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 6231 . . . 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 6437 . . 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 2651 . . . . . . 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 16906 . . . . . 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 16780 . . . . . 6 (𝜑 → ((𝐴𝑃)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))((𝑋(2nd𝐹)𝑃)‘𝐾)) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
227225, 226eqtrd 2685 . . . . 5 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
228227coeq1d 5316 . . . 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 16961 . . . . . . . 8 (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋)))
230229simprd 478 . . . . . . 7 (𝜑 → (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋))
231229simpld 474 . . . . . . . 8 (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
232172, 205, 207, 10, 54, 28evlf1 16907 . . . . . . . 8 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
233231, 211, 232feq123d 6072 . . . . . . 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 2651 . . . . . . 7 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋)))
236235fmpt 6421 . . . . . 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 6440 . . . . . 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 694 . . . . 5 (𝜑 → ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) = (𝑦 ∈ ((1st𝐹)‘𝑋) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦))))
240 fveq2 6229 . . . . . 6 (𝑦 = ((𝑎𝑋)‘( 1𝑋)) → (((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
241240fveq2d 6233 . . . . 5 (𝑦 = ((𝑎𝑋)‘( 1𝑋)) → ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦)) = ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋)))))
242237, 231, 239, 241fmptcof 6437 . . . 4 (𝜑 → (((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
243228, 242eqtrd 2685 . . 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 2695 . 2 (𝜑 → ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)))
245 eqid 2651 . . 3 (comp‘𝑇) = (comp‘𝑇)
246174simprd 478 . . . . . . 7 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
247 1st2ndbr 7261 . . . . . . 7 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
248168, 246, 247sylancr 696 . . . . . 6 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
249165, 194, 248funcf1 16573 . . . . 5 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
250249, 62, 27fovrnd 6848 . . . 4 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ (Base‘𝑇))
251250, 197eleqtrrd 2733 . . 3 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ 𝑉)
252219simprd 478 . . 3 (𝜑 → (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃))
253169, 19, 245, 198, 200, 251, 202, 252setcco 16780 . 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 6848 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ (Base‘𝑇))
255254, 197eleqtrrd 2733 . . 3 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ 𝑉)
256165, 166, 167, 248, 179, 181funcf2 16575 . . . . . 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 6693 . . . . . . . . . 10 (𝐹(1st𝐸)𝑋) = ((1st𝐸)‘⟨𝐹, 𝑋⟩)
258 df-ov 6693 . . . . . . . . . 10 (𝐺(1st𝐸)𝑃) = ((1st𝐸)‘⟨𝐺, 𝑃⟩)
259257, 258oveq12i 6702 . . . . . . . . 9 ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)) = (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩))
260259eqcomi 2660 . . . . . . . 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 6078 . . . . . 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 6848 . . . 4 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
265169, 19, 167, 255, 251elsetchom 16778 . . . 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 16780 . 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 2695 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 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cun 3605  wss 3607  cop 4216   class class class wbr 4685  cmpt 4762   × cxp 5141  ran crn 5144  ccom 5147  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  tpos ctpos 7396  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  Homf chomf 16374  oppCatcoppc 16418   Func cfunc 16561  func ccofu 16563   Nat cnat 16648   FuncCat cfuc 16649  SetCatcsetc 16772   ×c cxpc 16855   1stF c1stf 16856   2ndF c2ndf 16857   ⟨,⟩F cprf 16858   evalF cevlf 16896  HomFchof 16935  Yoncyon 16936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-homf 16378  df-comf 16379  df-oppc 16419  df-ssc 16517  df-resc 16518  df-subc 16519  df-func 16565  df-cofu 16567  df-nat 16650  df-fuc 16651  df-setc 16773  df-xpc 16859  df-1stf 16860  df-2ndf 16861  df-prf 16862  df-evlf 16900  df-curf 16901  df-hof 16937  df-yon 16938
This theorem is referenced by:  yonedalem3  16967
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