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Theorem yonedalem21 18250

Description: Lemma for yoneda 18260. (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 6894	. . . . 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 7427	. . . 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 7417	. . . 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 2755	. . 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 2727	. . . . 5 ⊢ (𝑄 ×_c 𝑂) = (𝑄 ×_c 𝑂)
7		yoneda.q	. . . . . 6 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8	7	fucbas 17936	. . . . 5 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
9		yoneda.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
10		yoneda.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	oppcbas 17684	. . . . 5 ⊢ 𝐵 = (Base‘𝑂)
12	6, 8, 11	xpcbas 18154	. . . 4 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×_c 𝑂))
13		eqid 2727	. . . . 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 2727	. . . . 5 ⊢ ((oppCat‘𝑄) ×_c 𝑄) = ((oppCat‘𝑄) ×_c 𝑄)
15		yoneda.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	9	oppccat 17689	. . . . . . . . 9 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
17	15, 16	syl 17	. . . . . . . 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 4184	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
21	18, 20	ssexd 5318	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
22		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
23	22	setccat 18059	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
24	21, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
25	7, 17, 24	fuccat 17947	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
26		eqid 2727	. . . . . . 7 ⊢ (𝑄 2^nd_F 𝑂) = (𝑄 2^nd_F 𝑂)
27	6, 25, 17, 26	2ndfcl 18174	. . . . . 6 ⊢ (𝜑 → (𝑄 2^nd_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑂))
28		eqid 2727	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
29		relfunc 17833	. . . . . . . . 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 18239	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
33		1st2ndbr 8038	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
34	29, 32, 33	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
35	9, 28, 34	funcoppc 17846	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2^nd ‘𝑌))
36		df-br 5143	. . . . . . 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 217	. . . . . 6 ⊢ (𝜑 → ⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
38	27, 37	cofucl 17859	. . . . 5 ⊢ (𝜑 → (⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ∈ ((𝑄 ×_c 𝑂) Func (oppCat‘𝑄)))
39		eqid 2727	. . . . . 6 ⊢ (𝑄 1^st_F 𝑂) = (𝑄 1^st_F 𝑂)
40	6, 25, 17, 39	1stfcl 18173	. . . . 5 ⊢ (𝜑 → (𝑄 1^st_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑄))
41	13, 14, 38, 40	prfcl 18179	. . . 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 4183	. . . . 5 ⊢ (𝜑 → ran (Hom_f ‘𝑄) ⊆ 𝑉)
45	42, 28, 43, 25, 18, 44	hofcl 18236	. . . 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 5711	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
49	12, 41, 45, 48	cofu1 17855	. . 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 2779	. 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 2727	. . . . . 6 ⊢ (Hom ‘(𝑄 ×_c 𝑂)) = (Hom ‘(𝑄 ×_c 𝑂))
52	13, 12, 51, 38, 40, 48	prf1 18176	. . . . 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 17855	. . . . . . 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 6904	. . . . . . . . . 10 ⊢ (1^st ‘𝑌) ∈ V
55		fvex 6904	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑌) ∈ V
56	55	tposex 8257	. . . . . . . . . 10 ⊢ tpos (2^nd ‘𝑌) ∈ V
57	54, 56	op1st 7993	. . . . . . . . 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 18171	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = (2^nd ‘⟨𝐹, 𝑋⟩))
60		op2ndg 7998	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
61	46, 47, 60	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
62	59, 61	eqtrd 2767	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
63	58, 62	fveq12d 6898	. . . . . . 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 2767	. . . . . 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 18168	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = (1^st ‘⟨𝐹, 𝑋⟩))
66		op1stg 7997	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
67	46, 47, 66	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
68	65, 67	eqtrd 2767	. . . . . 6 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
69	64, 68	opeq12d 4877	. . . . 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 2767	. . . 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 6895	. . 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 7417	. . 3 ⊢ (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = ((1^st ‘𝐻)‘⟨((1^st ‘𝑌)‘𝑋), 𝐹⟩)
73	71, 72	eqtr4di 2785	. 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 2727	. . . 4 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
75	7, 74	fuchom 17937	. . 3 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
76	30, 10, 15, 47, 9, 22, 21, 31	yon1cl 18240	. . 3 ⊢ (𝜑 → ((1^st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
77	42, 25, 8, 75, 76, 46	hof1 18231	. 2 ⊢ (𝜑 → (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
78	50, 73, 77	3eqtrd 2771	1 ⊢ (𝜑 → (𝐹(1^st ‘𝑍)𝑋) = (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∪ cun 3942 ⊆ wss 3944 ⟨cop 4630 class class class wbr 5142 × cxp 5670 ran crn 5673 Rel wrel 5677 ‘cfv 6542 (class class class)co 7414 1^st c1st 7983 2^nd c2nd 7984 tpos ctpos 8222 Basecbs 17165 Hom chom 17229 Catccat 17629 Idccid 17630 Hom_f chomf 17631 oppCatcoppc 17676 Func cfunc 17825 ∘_func ccofu 17827 Nat cnat 17916 FuncCat cfuc 17917 SetCatcsetc 18049 ×_c cxpc 18144 1^st_F c1stf 18145 2^nd_F c2ndf 18146 ⟨,⟩_F cprf 18147 eval_F cevlf 18186 Hom_Fchof 18225 Yoncyon 18226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-hom 17242 df-cco 17243 df-cat 17633 df-cid 17634 df-homf 17635 df-comf 17636 df-oppc 17677 df-func 17829 df-cofu 17831 df-nat 17918 df-fuc 17919 df-setc 18050 df-xpc 18148 df-1stf 18149 df-2ndf 18150 df-prf 18151 df-curf 18191 df-hof 18227 df-yon 18228

This theorem is referenced by: yonedalem3a 18251 yonedalem3b 18256 yonedainv 18258 yonffthlem 18259

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