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Theorem yonedalem3b 17127
Description: Lemma for yoneda 17131. (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 6885 . . . . . . . 8 (𝑏 = 𝑎 → (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑏) = (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎))
21oveq1d 6892 . . . . . . 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 6413 . . . . . 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 6413 . . . . 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 4951 . . . 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 2813 . . . . . . . . 9 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8 yoneda.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
9 yoneda.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
108, 9oppcbas 16585 . . . . . . . . 9 𝐵 = (Base‘𝑂)
11 eqid 2813 . . . . . . . . 9 (comp‘𝑆) = (comp‘𝑆)
12 eqid 2813 . . . . . . . . 9 (comp‘𝑄) = (comp‘𝑄)
13 eqid 2813 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
146, 7fuchom 16828 . . . . . . . . . . . 12 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
15 relfunc 16729 . . . . . . . . . . . . 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 3997 . . . . . . . . . . . . . . 15 (𝜑𝑈𝑉)
2219, 21ssexd 5007 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ V)
23 yoneda.u . . . . . . . . . . . . . 14 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2416, 17, 8, 18, 6, 22, 23yoncl 17110 . . . . . . . . . . . . 13 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
25 1st2ndbr 7452 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
2615, 24, 25sylancr 577 . . . . . . . . . . . 12 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
27 yonedalem22.p . . . . . . . . . . . 12 (𝜑𝑃𝐵)
28 yonedalem21.x . . . . . . . . . . . 12 (𝜑𝑋𝐵)
299, 13, 14, 26, 27, 28funcf2 16735 . . . . . . . . . . 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 6585 . . . . . . . . . 10 (𝜑 → ((𝑃(2nd𝑌)𝑋)‘𝐾) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st𝑌)‘𝑋)))
3231adantr 468 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑃(2nd𝑌)𝑋)‘𝐾) ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)((1st𝑌)‘𝑋)))
33 simpr 473 . . . . . . . . . 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 468 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
366, 7, 12, 33, 35fuccocl 16831 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐺))
3727adantr 468 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑃𝐵)
386, 7, 10, 11, 12, 32, 36, 37fuccoval 16830 . . . . . . . 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 16830 . . . . . . . . . 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 468 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
41 eqid 2813 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
42 relfunc 16729 . . . . . . . . . . . . . . . 16 Rel (𝑂 Func 𝑆)
436fucbas 16827 . . . . . . . . . . . . . . . . . 18 (𝑂 Func 𝑆) = (Base‘𝑄)
449, 43, 26funcf1 16733 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
4544, 28ffvelrnd 6585 . . . . . . . . . . . . . . . 16 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
46 1st2ndbr 7452 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
4742, 45, 46sylancr 577 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
4810, 41, 47funcf1 16733 . . . . . . . . . . . . . 14 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
4918, 22setcbas 16935 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝑆))
5049feq3d 6246 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
5148, 50mpbird 248 . . . . . . . . . . . . 13 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
5251, 27ffvelrnd 6585 . . . . . . . . . . . 12 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∈ 𝑈)
5352adantr 468 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑃) ∈ 𝑈)
54 yonedalem21.f . . . . . . . . . . . . . . . 16 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
55 1st2ndbr 7452 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
5642, 54, 55sylancr 577 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
5710, 41, 56funcf1 16733 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
5849feq3d 6246 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹):𝐵𝑈 ↔ (1st𝐹):𝐵⟶(Base‘𝑆)))
5957, 58mpbird 248 . . . . . . . . . . . . 13 (𝜑 → (1st𝐹):𝐵𝑈)
6059, 27ffvelrnd 6585 . . . . . . . . . . . 12 (𝜑 → ((1st𝐹)‘𝑃) ∈ 𝑈)
6160adantr 468 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑃) ∈ 𝑈)
62 yonedalem22.g . . . . . . . . . . . . . . . 16 (𝜑𝐺 ∈ (𝑂 Func 𝑆))
63 1st2ndbr 7452 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐺 ∈ (𝑂 Func 𝑆)) → (1st𝐺)(𝑂 Func 𝑆)(2nd𝐺))
6442, 62, 63sylancr 577 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝐺)(𝑂 Func 𝑆)(2nd𝐺))
6510, 41, 64funcf1 16733 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝑆))
6665, 27ffvelrnd 6585 . . . . . . . . . . . . 13 (𝜑 → ((1st𝐺)‘𝑃) ∈ (Base‘𝑆))
6766, 49eleqtrrd 2895 . . . . . . . . . . . 12 (𝜑 → ((1st𝐺)‘𝑃) ∈ 𝑈)
6867adantr 468 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐺)‘𝑃) ∈ 𝑈)
697, 33nat1st2nd 16818 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (⟨(1st ‘((1st𝑌)‘𝑋)), (2nd ‘((1st𝑌)‘𝑋))⟩(𝑂 Nat 𝑆)⟨(1st𝐹), (2nd𝐹)⟩))
70 eqid 2813 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
717, 69, 10, 70, 37natcl 16820 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑃) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑃)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
7218, 40, 70, 53, 61elsetchom 16938 . . . . . . . . . . . 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 223 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃))
747, 34nat1st2nd 16818 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩(𝑂 Nat 𝑆)⟨(1st𝐺), (2nd𝐺)⟩))
757, 74, 10, 70, 27natcl 16820 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝑃) ∈ (((1st𝐹)‘𝑃)(Hom ‘𝑆)((1st𝐺)‘𝑃)))
7618, 22, 70, 60, 67elsetchom 16938 . . . . . . . . . . . . 13 (𝜑 → ((𝐴𝑃) ∈ (((1st𝐹)‘𝑃)(Hom ‘𝑆)((1st𝐺)‘𝑃)) ↔ (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃)))
7775, 76mpbid 223 . . . . . . . . . . . 12 (𝜑 → (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃))
7877adantr 468 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃))
7918, 40, 11, 53, 61, 68, 73, 78setcco 16940 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑃), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))(𝑎𝑃)) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8039, 79eqtrd 2847 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴(⟨((1st𝑌)‘𝑋), 𝐹⟩(comp‘𝑄)𝐺)𝑎)‘𝑃) = ((𝐴𝑃) ∘ (𝑎𝑃)))
8180oveq1d 6892 . . . . . . . 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 6585 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝑌)‘𝑃) ∈ (𝑂 Func 𝑆))
83 1st2ndbr 7452 . . . . . . . . . . . . . 14 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑃) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑃)))
8442, 82, 83sylancr 577 . . . . . . . . . . . . 13 (𝜑 → (1st ‘((1st𝑌)‘𝑃))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑃)))
8510, 41, 84funcf1 16733 . . . . . . . . . . . 12 (𝜑 → (1st ‘((1st𝑌)‘𝑃)):𝐵⟶(Base‘𝑆))
8685, 27ffvelrnd 6585 . . . . . . . . . . 11 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ (Base‘𝑆))
8786, 49eleqtrrd 2895 . . . . . . . . . 10 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
8887adantr 468 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) ∈ 𝑈)
897, 31nat1st2nd 16818 . . . . . . . . . . . 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 16820 . . . . . . . . . . 11 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃) ∈ (((1st ‘((1st𝑌)‘𝑃))‘𝑃)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
9118, 22, 70, 87, 52elsetchom 16938 . . . . . . . . . . 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 223 . . . . . . . . . 10 (𝜑 → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
9392adantr 468 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃):((1st ‘((1st𝑌)‘𝑃))‘𝑃)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
94 fco 6276 . . . . . . . . . 10 (((𝐴𝑃):((1st𝐹)‘𝑃)⟶((1st𝐺)‘𝑃) ∧ (𝑎𝑃):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐹)‘𝑃)) → ((𝐴𝑃) ∘ (𝑎𝑃)):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐺)‘𝑃))
9578, 73, 94syl2anc 575 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝐴𝑃) ∘ (𝑎𝑃)):((1st ‘((1st𝑌)‘𝑋))‘𝑃)⟶((1st𝐺)‘𝑃))
9618, 40, 11, 88, 53, 68, 93, 95setcco 16940 . . . . . . . 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 2851 . . . . . . 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 6413 . . . . . 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 16550 . . . . . . . . 9 (𝜑 → ( 1𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃))
10116, 9, 17, 27, 13, 27yon11 17112 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑃))‘𝑃) = (𝑃(Hom ‘𝐶)𝑃))
102100, 101eleqtrrd 2895 . . . . . . . 8 (𝜑 → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
103102adantr 468 . . . . . . 7 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑃) ∈ ((1st ‘((1st𝑌)‘𝑃))‘𝑃))
104 fvco3 6499 . . . . . . 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 575 . . . . . 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 6585 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑃))
107 fvco3 6499 . . . . . . . 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 575 . . . . . . 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 468 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐶 ∈ Cat)
11028adantr 468 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋𝐵)
111 eqid 2813 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
11230adantr 468 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))
113100adantr 468 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑃) ∈ (𝑃(Hom ‘𝐶)𝑃))
11416, 9, 109, 37, 13, 110, 111, 37, 112, 113yon2 17114 . . . . . . . . . . 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 16552 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝐾(⟨𝑃, 𝑃⟩(comp‘𝐶)𝑋)( 1𝑃)) = 𝐾)
116114, 115eqtrd 2847 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃)) = 𝐾)
117116fveq2d 6415 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = ((𝑎𝑃)‘𝐾))
118 eqid 2813 . . . . . . . . . . . . . . 15 (Hom ‘𝑂) = (Hom ‘𝑂)
11910, 118, 70, 47, 28, 27funcf2 16735 . . . . . . . . . . . . . 14 (𝜑 → (𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12013, 8oppchom 16582 . . . . . . . . . . . . . . 15 (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
12130, 120syl6eleqr 2903 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
122119, 121ffvelrnd 6585 . . . . . . . . . . . . 13 (𝜑 → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑃)))
12351, 28ffvelrnd 6585 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
12418, 22, 70, 123, 52elsetchom 16938 . . . . . . . . . . . . 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 223 . . . . . . . . . . . 12 (𝜑 → ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑃))
126125adantr 468 . . . . . . . . . . 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 16550 . . . . . . . . . . . . 13 (𝜑 → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
12816, 9, 17, 28, 13, 28yon11 17112 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
129127, 128eleqtrrd 2895 . . . . . . . . . . . 12 (𝜑 → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
130129adantr 468 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑋))
131 fvco3 6499 . . . . . . . . . . 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 575 . . . . . . . . . 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 468 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
1347, 69, 10, 118, 11, 110, 37, 133nati 16822 . . . . . . . . . . . 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 468 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
13618, 40, 11, 135, 53, 61, 126, 73setcco 16940 . . . . . . . . . . . 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 6585 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
138137adantr 468 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
1397, 69, 10, 70, 110natcl 16820 . . . . . . . . . . . . . 14 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑋)))
14018, 40, 70, 135, 138elsetchom 16938 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎𝑋):((1st ‘((1st𝑌)‘𝑋))‘𝑋)⟶((1st𝐹)‘𝑋))
14210, 118, 70, 56, 28, 27funcf2 16735 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋(2nd𝐹)𝑃):(𝑋(Hom ‘𝑂)𝑃)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
143142, 121ffvelrnd 6585 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑋(2nd𝐹)𝑃)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)))
14418, 22, 70, 137, 60elsetchom 16938 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑋(2nd𝐹)𝑃)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑃)) ↔ ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃)))
145143, 144mpbid 223 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃))
146145adantr 468 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑋(2nd𝐹)𝑃)‘𝐾):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑃))
14718, 40, 11, 135, 138, 61, 141, 146setcco 16940 . . . . . . . . . . . 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 2855 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)) = (((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋)))
149148fveq1d 6413 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑎𝑃) ∘ ((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾))‘( 1𝑋)) = ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)))
150127adantr 468 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
15116, 9, 109, 110, 13, 110, 111, 37, 112, 150yon12 17113 . . . . . . . . . . . 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 16551 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (( 1𝑋)(⟨𝑃, 𝑋⟩(comp‘𝐶)𝑋)𝐾) = 𝐾)
153151, 152eqtrd 2847 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋)) = 𝐾)
154153fveq2d 6415 . . . . . . . . . 10 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘(((𝑋(2nd ‘((1st𝑌)‘𝑋))𝑃)‘𝐾)‘( 1𝑋))) = ((𝑎𝑃)‘𝐾))
155132, 149, 1543eqtr3d 2855 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = ((𝑎𝑃)‘𝐾))
156 fvco3 6499 . . . . . . . . . 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 575 . . . . . . . . 9 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((((𝑋(2nd𝐹)𝑃)‘𝐾) ∘ (𝑎𝑋))‘( 1𝑋)) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
158117, 155, 1573eqtr2d 2853 . . . . . . . 8 ((𝜑𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎𝑃)‘((((𝑃(2nd𝑌)𝑋)‘𝐾)‘𝑃)‘( 1𝑃))) = (((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))
159158fveq2d 6415 . . . . . . 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 2847 . . . . . 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 2851 . . . . 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 4945 . . . 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 2859 . . 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 2813 . . . . . . . . . . 11 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
165164, 43, 10xpcbas 17026 . . . . . . . . . 10 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
166 eqid 2813 . . . . . . . . . 10 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
167 eqid 2813 . . . . . . . . . 10 (Hom ‘𝑇) = (Hom ‘𝑇)
168 relfunc 16729 . . . . . . . . . . 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 17120 . . . . . . . . . . . 12 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
175174simpld 484 . . . . . . . . . . 11 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
176 1st2ndbr 7452 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
177168, 175, 176sylancr 577 . . . . . . . . . 10 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
178 opelxpi 5355 . . . . . . . . . . 11 ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋𝐵) → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
17954, 28, 178syl2anc 575 . . . . . . . . . 10 (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
180 opelxpi 5355 . . . . . . . . . . 11 ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃𝐵) → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
18162, 27, 180syl2anc 575 . . . . . . . . . 10 (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
182165, 166, 167, 177, 179, 181funcf2 16735 . . . . . . . . 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 17034 . . . . . . . . . . 11 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
184120xpeq2i 5344 . . . . . . . . . . 11 ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))
185183, 184syl6eq 2863 . . . . . . . . . 10 (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋)))
186 df-ov 6880 . . . . . . . . . . . . 13 (𝐹(1st𝑍)𝑋) = ((1st𝑍)‘⟨𝐹, 𝑋⟩)
187 df-ov 6880 . . . . . . . . . . . . 13 (𝐺(1st𝑍)𝑃) = ((1st𝑍)‘⟨𝐺, 𝑃⟩)
188186, 187oveq12i 6889 . . . . . . . . . . . 12 ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)) = (((1st𝑍)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝑍)‘⟨𝐺, 𝑃⟩))
189188eqcomi 2822 . . . . . . . . . . 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 6254 . . . . . . . . 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 223 . . . . . . . 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 7039 . . . . . . 7 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝑍)𝑋)(Hom ‘𝑇)(𝐺(1st𝑍)𝑃)))
194 eqid 2813 . . . . . . . . . . 11 (Base‘𝑇) = (Base‘𝑇)
195165, 194, 177funcf1 16733 . . . . . . . . . 10 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
196195, 54, 28fovrnd 7039 . . . . . . . . 9 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ (Base‘𝑇))
197169, 19setcbas 16935 . . . . . . . . 9 (𝜑𝑉 = (Base‘𝑇))
198196, 197eleqtrrd 2895 . . . . . . . 8 (𝜑 → (𝐹(1st𝑍)𝑋) ∈ 𝑉)
199195, 62, 27fovrnd 7039 . . . . . . . . 9 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ (Base‘𝑇))
200199, 197eleqtrrd 2895 . . . . . . . 8 (𝜑 → (𝐺(1st𝑍)𝑃) ∈ 𝑉)
201169, 19, 167, 198, 200elsetchom 16938 . . . . . . 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 223 . . . . . 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 17126 . . . . . . . 8 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))
2048oppccat 16589 . . . . . . . . . . 11 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
20517, 204syl 17 . . . . . . . . . 10 (𝜑𝑂 ∈ Cat)
20618setccat 16942 . . . . . . . . . . 11 (𝑈 ∈ V → 𝑆 ∈ Cat)
20722, 206syl 17 . . . . . . . . . 10 (𝜑𝑆 ∈ Cat)
2086, 205, 207fuccat 16837 . . . . . . . . 9 (𝜑𝑄 ∈ Cat)
209170, 208, 43, 14, 45, 54, 82, 62, 12, 31, 34hof2val 17104 . . . . . . . 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 2847 . . . . . . 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 17121 . . . . . . 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 17121 . . . . . . 7 (𝜑 → (𝐺(1st𝑍)𝑃) = (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺))
213210, 211, 212feq123d 6248 . . . . . 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 223 . . . . 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 2813 . . . . . 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 6605 . . . . 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 225 . . . 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 17122 . . . . 5 (𝜑 → ((𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))) ∧ (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃)))
220219simpld 484 . . . 4 (𝜑 → (𝐺𝑀𝑃) = (𝑎 ∈ (((1st𝑌)‘𝑃)(𝑂 Nat 𝑆)𝐺) ↦ ((𝑎𝑃)‘( 1𝑃))))
221 fveq1 6410 . . . . 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 6413 . . . 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 6623 . . 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 2813 . . . . . . 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 17067 . . . . . 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 16940 . . . . . 6 (𝜑 → ((𝐴𝑃)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑃)⟩(comp‘𝑆)((1st𝐺)‘𝑃))((𝑋(2nd𝐹)𝑃)‘𝐾)) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
227225, 226eqtrd 2847 . . . . 5 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) = ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)))
228227coeq1d 5492 . . . 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 17122 . . . . . . . 8 (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋)))
230229simprd 485 . . . . . . 7 (𝜑 → (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋))
231229simpld 484 . . . . . . . 8 (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))))
232172, 205, 207, 10, 54, 28evlf1 17068 . . . . . . . 8 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
233231, 211, 232feq123d 6248 . . . . . . 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 223 . . . . . 6 (𝜑 → (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))):(((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st𝐹)‘𝑋))
235 eqid 2813 . . . . . . 7 (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋)))
236235fmpt 6605 . . . . . 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 225 . . . . 5 (𝜑 → ∀𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)((𝑎𝑋)‘( 1𝑋)) ∈ ((1st𝐹)‘𝑋))
238 fcompt 6626 . . . . . 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 575 . . . . 5 (𝜑 → ((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) = (𝑦 ∈ ((1st𝐹)‘𝑋) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦))))
240 2fveq3 6416 . . . . 5 (𝑦 = ((𝑎𝑋)‘( 1𝑋)) → ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘𝑦)) = ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋)))))
241237, 231, 239, 240fmptcof 6623 . . . 4 (𝜑 → (((𝐴𝑃) ∘ ((𝑋(2nd𝐹)𝑃)‘𝐾)) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
242228, 241eqtrd 2847 . . 3 (𝜑 → ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴𝑃)‘(((𝑋(2nd𝐹)𝑃)‘𝐾)‘((𝑎𝑋)‘( 1𝑋))))))
243163, 223, 2423eqtr4d 2857 . 2 (𝜑 → ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∘ (𝐹𝑀𝑋)))
244 eqid 2813 . . 3 (comp‘𝑇) = (comp‘𝑇)
245174simprd 485 . . . . . . 7 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
246 1st2ndbr 7452 . . . . . . 7 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
247168, 245, 246sylancr 577 . . . . . 6 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
248165, 194, 247funcf1 16733 . . . . 5 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
249248, 62, 27fovrnd 7039 . . . 4 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ (Base‘𝑇))
250249, 197eleqtrrd 2895 . . 3 (𝜑 → (𝐺(1st𝐸)𝑃) ∈ 𝑉)
251219simprd 485 . . 3 (𝜑 → (𝐺𝑀𝑃):(𝐺(1st𝑍)𝑃)⟶(𝐺(1st𝐸)𝑃))
252169, 19, 244, 198, 200, 250, 202, 251setcco 16940 . 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)))
253248, 54, 28fovrnd 7039 . . . 4 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ (Base‘𝑇))
254253, 197eleqtrrd 2895 . . 3 (𝜑 → (𝐹(1st𝐸)𝑋) ∈ 𝑉)
255165, 166, 167, 247, 179, 181funcf2 16735 . . . . . 6 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩):(⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)⟶(((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)))
256 df-ov 6880 . . . . . . . . . 10 (𝐹(1st𝐸)𝑋) = ((1st𝐸)‘⟨𝐹, 𝑋⟩)
257 df-ov 6880 . . . . . . . . . 10 (𝐺(1st𝐸)𝑃) = ((1st𝐸)‘⟨𝐺, 𝑃⟩)
258256, 257oveq12i 6889 . . . . . . . . 9 ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)) = (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩))
259258eqcomi 2822 . . . . . . . 8 (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃))
260259a1i 11 . . . . . . 7 (𝜑 → (((1st𝐸)‘⟨𝐹, 𝑋⟩)(Hom ‘𝑇)((1st𝐸)‘⟨𝐺, 𝑃⟩)) = ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
261185, 260feq23d 6254 . . . . . 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))))
262255, 261mpbid 223 . . . . 5 (𝜑 → (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩):((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑃(Hom ‘𝐶)𝑋))⟶((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
263262, 34, 30fovrnd 7039 . . . 4 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾) ∈ ((𝐹(1st𝐸)𝑋)(Hom ‘𝑇)(𝐺(1st𝐸)𝑃)))
264169, 19, 167, 254, 250elsetchom 16938 . . . 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)))
265263, 264mpbid 223 . . 3 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾):(𝐹(1st𝐸)𝑋)⟶(𝐺(1st𝐸)𝑃))
266169, 19, 244, 198, 254, 250, 230, 265setcco 16940 . 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)))
267243, 252, 2663eqtr4d 2857 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 384   = wceq 1637  wcel 2157  wral 3103  Vcvv 3398  cun 3774  wss 3776  cop 4383   class class class wbr 4851  cmpt 4930   × cxp 5316  ran crn 5319  ccom 5322  Rel wrel 5323  wf 6100  cfv 6104  (class class class)co 6877  cmpt2 6879  1st c1st 7399  2nd c2nd 7400  tpos ctpos 7589  Basecbs 16071  Hom chom 16167  compcco 16168  Catccat 16532  Idccid 16533  Homf chomf 16534  oppCatcoppc 16578   Func cfunc 16721  func ccofu 16723   Nat cnat 16808   FuncCat cfuc 16809  SetCatcsetc 16932   ×c cxpc 17016   1stF c1stf 17017   2ndF c2ndf 17018   ⟨,⟩F cprf 17019   evalF cevlf 17057  HomFchof 17096  Yoncyon 17097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-tpos 7590  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-ixp 8149  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-nn 11309  df-2 11367  df-3 11368  df-4 11369  df-5 11370  df-6 11371  df-7 11372  df-8 11373  df-9 11374  df-n0 11563  df-z 11647  df-dec 11763  df-uz 11908  df-fz 12553  df-struct 16073  df-ndx 16074  df-slot 16075  df-base 16077  df-sets 16078  df-ress 16079  df-hom 16180  df-cco 16181  df-cat 16536  df-cid 16537  df-homf 16538  df-comf 16539  df-oppc 16579  df-ssc 16677  df-resc 16678  df-subc 16679  df-func 16725  df-cofu 16727  df-nat 16810  df-fuc 16811  df-setc 16933  df-xpc 17020  df-1stf 17021  df-2ndf 17022  df-prf 17023  df-evlf 17061  df-curf 17062  df-hof 17098  df-yon 17099
This theorem is referenced by:  yonedalem3  17128
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