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Theorem yonedalem21 18473

Description: Lemma for yoneda 18483. (Contributed by Mario Carneiro, 28-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 = (Hom_F‘𝑄)
yoneda.r	⊢ 𝑅 = ((𝑄 ×_c 𝑂) FuncCat 𝑇)
yoneda.e	⊢ 𝐸 = (𝑂 eval_F 𝑆)
yoneda.z	⊢ 𝑍 = (𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))
yoneda.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yoneda.w	⊢ (𝜑 → 𝑉 ∈ 𝑊)
yoneda.u	⊢ (𝜑 → ran (Hom_f ‘𝐶) ⊆ 𝑈)
yoneda.v	⊢ (𝜑 → (ran (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
yonedalem21.f	⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
yonedalem21	⊢ (𝜑 → (𝐹(1^st ‘𝑍)𝑋) = (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

**Proof of Theorem yonedalem21**
Step	Hyp	Ref	Expression
1		yoneda.z	. . . . . 6 ⊢ 𝑍 = (𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))
2	1	fveq2i 6908	. . . . 5 ⊢ (1^st ‘𝑍) = (1^st ‘(𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂))))
3	2	oveqi 7453	. . . 4 ⊢ (𝐹(1^st ‘𝑍)𝑋) = (𝐹(1^st ‘(𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂))))𝑋)
4		df-ov 7443	. . . 4 ⊢ (𝐹(1^st ‘(𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂))))𝑋) = ((1^st ‘(𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂))))‘⟨𝐹, 𝑋⟩)
5	3, 4	eqtri 2786	. . 3 ⊢ (𝐹(1^st ‘𝑍)𝑋) = ((1^st ‘(𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂))))‘⟨𝐹, 𝑋⟩)
6		eqid 2763	. . . . 5 ⊢ (𝑄 ×_c 𝑂) = (𝑄 ×_c 𝑂)
7		yoneda.q	. . . . . 6 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8	7	fucbas 18164	. . . . 5 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
9		yoneda.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
10		yoneda.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	oppcbas 17918	. . . . 5 ⊢ 𝐵 = (Base‘𝑂)
12	6, 8, 11	xpcbas 18378	. . . 4 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×_c 𝑂))
13		eqid 2763	. . . . 5 ⊢ ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)) = ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂))
14		eqid 2763	. . . . 5 ⊢ ((oppCat‘𝑄) ×_c 𝑄) = ((oppCat‘𝑄) ×_c 𝑄)
15		yoneda.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	9	oppccat 17922	. . . . . . . . 9 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
17	15, 16	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ Cat)
18		yoneda.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
19		yoneda.v	. . . . . . . . . . 11 ⊢ (𝜑 → (ran (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
20	19	unssbd 4195	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
21	18, 20	ssexd 5290	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
22		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
23	22	setccat 18286	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
24	21, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
25	7, 17, 24	fuccat 18174	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
26		eqid 2763	. . . . . . 7 ⊢ (𝑄 2^nd_F 𝑂) = (𝑄 2^nd_F 𝑂)
27	6, 25, 17, 26	2ndfcl 18398	. . . . . 6 ⊢ (𝜑 → (𝑄 2^nd_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑂))
28		eqid 2763	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
29		relfunc 18063	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝑄)
30		yoneda.y	. . . . . . . . . 10 ⊢ 𝑌 = (Yon‘𝐶)
31		yoneda.u	. . . . . . . . . 10 ⊢ (𝜑 → ran (Hom_f ‘𝐶) ⊆ 𝑈)
32	30, 15, 9, 22, 7, 21, 31	yoncl 18462	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
33		1st2ndbr 8074	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
34	29, 32, 33	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
35	9, 28, 34	funcoppc 18076	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2^nd ‘𝑌))
36		df-br 5107	. . . . . . 7 ⊢ ((1^st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2^nd ‘𝑌) ↔ ⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
37	35, 36	sylib 221	. . . . . 6 ⊢ (𝜑 → ⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
38	27, 37	cofucl 18089	. . . . 5 ⊢ (𝜑 → (⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ∈ ((𝑄 ×_c 𝑂) Func (oppCat‘𝑄)))
39		eqid 2763	. . . . . 6 ⊢ (𝑄 1^st_F 𝑂) = (𝑄 1^st_F 𝑂)
40	6, 25, 17, 39	1stfcl 18397	. . . . 5 ⊢ (𝜑 → (𝑄 1^st_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑄))
41	13, 14, 38, 40	prfcl 18403	. . . 4 ⊢ (𝜑 → ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)) ∈ ((𝑄 ×_c 𝑂) Func ((oppCat‘𝑄) ×_c 𝑄)))
42		yoneda.h	. . . . 5 ⊢ 𝐻 = (Hom_F‘𝑄)
43		yoneda.t	. . . . 5 ⊢ 𝑇 = (SetCat‘𝑉)
44	19	unssad 4194	. . . . 5 ⊢ (𝜑 → ran (Hom_f ‘𝑄) ⊆ 𝑉)
45	42, 28, 43, 25, 18, 44	hofcl 18459	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×_c 𝑄) Func 𝑇))
46		yonedalem21.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
47		yonedalem21.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
48	46, 47	opelxpd 5694	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
49	12, 41, 45, 48	cofu1 18085	. . 3 ⊢ (𝜑 → ((1^st ‘(𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂))))‘⟨𝐹, 𝑋⟩) = ((1^st ‘𝐻)‘((1^st ‘((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))‘⟨𝐹, 𝑋⟩)))
50	5, 49	eqtrid 2810	. 2 ⊢ (𝜑 → (𝐹(1^st ‘𝑍)𝑋) = ((1^st ‘𝐻)‘((1^st ‘((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))‘⟨𝐹, 𝑋⟩)))
51		eqid 2763	. . . . . 6 ⊢ (Hom ‘(𝑄 ×_c 𝑂)) = (Hom ‘(𝑄 ×_c 𝑂))
52	13, 12, 51, 38, 40, 48	prf1 18400	. . . . 5 ⊢ (𝜑 → ((1^st ‘((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1^st ‘(⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)))‘⟨𝐹, 𝑋⟩), ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩)⟩)
53	12, 27, 37, 48	cofu1 18085	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1^st ‘⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩)‘((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩)))
54		fvex 6918	. . . . . . . . . 10 ⊢ (1^st ‘𝑌) ∈ V
55		fvex 6918	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑌) ∈ V
56	55	tposex 8302	. . . . . . . . . 10 ⊢ tpos (2^nd ‘𝑌) ∈ V
57	54, 56	op1st 8030	. . . . . . . . 9 ⊢ (1^st ‘⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩) = (1^st ‘𝑌)
58	57	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩) = (1^st ‘𝑌))
59	6, 12, 51, 25, 17, 26, 48	2ndf1 18395	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = (2^nd ‘⟨𝐹, 𝑋⟩))
60		op2ndg 8035	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
61	46, 47, 60	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
62	59, 61	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
63	58, 62	fveq12d 6912	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩)‘((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1^st ‘𝑌)‘𝑋))
64	53, 63	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → ((1^st ‘(⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1^st ‘𝑌)‘𝑋))
65	6, 12, 51, 25, 17, 39, 48	1stf1 18392	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = (1^st ‘⟨𝐹, 𝑋⟩))
66		op1stg 8034	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
67	46, 47, 66	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
68	65, 67	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
69	64, 68	opeq12d 4851	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘(⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)))‘⟨𝐹, 𝑋⟩), ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1^st ‘𝑌)‘𝑋), 𝐹⟩)
70	52, 69	eqtrd 2798	. . . 4 ⊢ (𝜑 → ((1^st ‘((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1^st ‘𝑌)‘𝑋), 𝐹⟩)
71	70	fveq2d 6909	. . 3 ⊢ (𝜑 → ((1^st ‘𝐻)‘((1^st ‘((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))‘⟨𝐹, 𝑋⟩)) = ((1^st ‘𝐻)‘⟨((1^st ‘𝑌)‘𝑋), 𝐹⟩))
72		df-ov 7443	. . 3 ⊢ (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = ((1^st ‘𝐻)‘⟨((1^st ‘𝑌)‘𝑋), 𝐹⟩)
73	71, 72	eqtr4di 2816	. 2 ⊢ (𝜑 → ((1^st ‘𝐻)‘((1^st ‘((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))‘⟨𝐹, 𝑋⟩)) = (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹))
74		eqid 2763	. . . 4 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
75	7, 74	fuchom 18165	. . 3 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
76	30, 10, 15, 47, 9, 22, 21, 31	yon1cl 18463	. . 3 ⊢ (𝜑 → ((1^st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
77	42, 25, 8, 75, 76, 46	hof1 18454	. 2 ⊢ (𝜑 → (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
78	50, 73, 77	3eqtrd 2802	1 ⊢ (𝜑 → (𝐹(1^st ‘𝑍)𝑋) = (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⊆ wss 3900 ∪ cun 4152 ⟨cop 4838 class class class wbr 5106 × cxp 5653 ran crn 5656 Rel wrel 5660 ‘cfv 6555 (class class class)co 7440 1^st c1st 8020 2^nd c2nd 8021 tpos ctpos 8267 Basecbs 17413 Hom chom 17465 Catccat 17864 Idccid 17865 Hom_f chomf 17866 oppCatcoppc 17911 Func cfunc 18055 ∘_func ccofu 18057 Nat cnat 18145 FuncCat cfuc 18146 SetCatcsetc 18276 ×_c cxpc 18368 1^st_F c1stf 18369 2^nd_F c2ndf 18370 ⟨,⟩_F cprf 18371 eval_F cevlf 18409 Hom_Fchof 18448 Yoncyon 18449

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-tpos 8268 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-er 8742 df-map 8874 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-fz 13664 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-hom 17478 df-cco 17479 df-cat 17868 df-cid 17869 df-homf 17870 df-comf 17871 df-oppc 17912 df-func 18059 df-cofu 18061 df-nat 18147 df-fuc 18148 df-setc 18277 df-xpc 18372 df-1stf 18373 df-2ndf 18374 df-prf 18375 df-curf 18414 df-hof 18450 df-yon 18451

This theorem is used by: yonedalem3a 18474 yonedalem3b 18479 yonedainv 18481 yonffthlem 18482

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