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Theorem yonedalem21 18259

Description: Lemma for yoneda 18269. (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 6893	. . . . 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 7426	. . . 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 7416	. . . 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 2753	. . 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 2725	. . . . 5 ⊢ (𝑄 ×_c 𝑂) = (𝑄 ×_c 𝑂)
7		yoneda.q	. . . . . 6 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8	7	fucbas 17945	. . . . 5 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
9		yoneda.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
10		yoneda.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	oppcbas 17693	. . . . 5 ⊢ 𝐵 = (Base‘𝑂)
12	6, 8, 11	xpcbas 18163	. . . 4 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×_c 𝑂))
13		eqid 2725	. . . . 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 2725	. . . . 5 ⊢ ((oppCat‘𝑄) ×_c 𝑄) = ((oppCat‘𝑄) ×_c 𝑄)
15		yoneda.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	9	oppccat 17698	. . . . . . . . 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 4183	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
21	18, 20	ssexd 5320	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
22		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
23	22	setccat 18068	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
24	21, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
25	7, 17, 24	fuccat 17956	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
26		eqid 2725	. . . . . . 7 ⊢ (𝑄 2^nd_F 𝑂) = (𝑄 2^nd_F 𝑂)
27	6, 25, 17, 26	2ndfcl 18183	. . . . . 6 ⊢ (𝜑 → (𝑄 2^nd_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑂))
28		eqid 2725	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
29		relfunc 17842	. . . . . . . . 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 18248	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
33		1st2ndbr 8040	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
34	29, 32, 33	sylancr 585	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
35	9, 28, 34	funcoppc 17855	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2^nd ‘𝑌))
36		df-br 5145	. . . . . . 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 17868	. . . . 5 ⊢ (𝜑 → (⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ∈ ((𝑄 ×_c 𝑂) Func (oppCat‘𝑄)))
39		eqid 2725	. . . . . 6 ⊢ (𝑄 1^st_F 𝑂) = (𝑄 1^st_F 𝑂)
40	6, 25, 17, 39	1stfcl 18182	. . . . 5 ⊢ (𝜑 → (𝑄 1^st_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑄))
41	13, 14, 38, 40	prfcl 18188	. . . 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 4182	. . . . 5 ⊢ (𝜑 → ran (Hom_f ‘𝑄) ⊆ 𝑉)
45	42, 28, 43, 25, 18, 44	hofcl 18245	. . . 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 5712	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
49	12, 41, 45, 48	cofu1 17864	. . 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 2777	. 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 2725	. . . . . 6 ⊢ (Hom ‘(𝑄 ×_c 𝑂)) = (Hom ‘(𝑄 ×_c 𝑂))
52	13, 12, 51, 38, 40, 48	prf1 18185	. . . . 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 17864	. . . . . . 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 6903	. . . . . . . . . 10 ⊢ (1^st ‘𝑌) ∈ V
55		fvex 6903	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑌) ∈ V
56	55	tposex 8259	. . . . . . . . . 10 ⊢ tpos (2^nd ‘𝑌) ∈ V
57	54, 56	op1st 7995	. . . . . . . . 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 18180	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = (2^nd ‘⟨𝐹, 𝑋⟩))
60		op2ndg 8000	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
61	46, 47, 60	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
62	59, 61	eqtrd 2765	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
63	58, 62	fveq12d 6897	. . . . . . 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 2765	. . . . . 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 18177	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = (1^st ‘⟨𝐹, 𝑋⟩))
66		op1stg 7999	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
67	46, 47, 66	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
68	65, 67	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
69	64, 68	opeq12d 4878	. . . . 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 2765	. . . 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 6894	. . 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 7416	. . 3 ⊢ (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = ((1^st ‘𝐻)‘⟨((1^st ‘𝑌)‘𝑋), 𝐹⟩)
73	71, 72	eqtr4di 2783	. 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 2725	. . . 4 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
75	7, 74	fuchom 17946	. . 3 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
76	30, 10, 15, 47, 9, 22, 21, 31	yon1cl 18249	. . 3 ⊢ (𝜑 → ((1^st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
77	42, 25, 8, 75, 76, 46	hof1 18240	. 2 ⊢ (𝜑 → (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
78	50, 73, 77	3eqtrd 2769	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 1533 ∈ wcel 2098 Vcvv 3463 ∪ cun 3939 ⊆ wss 3941 ⟨cop 4631 class class class wbr 5144 × cxp 5671 ran crn 5674 Rel wrel 5678 ‘cfv 6543 (class class class)co 7413 1^st c1st 7985 2^nd c2nd 7986 tpos ctpos 8224 Basecbs 17174 Hom chom 17238 Catccat 17638 Idccid 17639 Hom_f chomf 17640 oppCatcoppc 17685 Func cfunc 17834 ∘_func ccofu 17836 Nat cnat 17925 FuncCat cfuc 17926 SetCatcsetc 18058 ×_c cxpc 18153 1^st_F c1stf 18154 2^nd_F c2ndf 18155 ⟨,⟩_F cprf 18156 eval_F cevlf 18195 Hom_Fchof 18234 Yoncyon 18235

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-hom 17251 df-cco 17252 df-cat 17642 df-cid 17643 df-homf 17644 df-comf 17645 df-oppc 17686 df-func 17838 df-cofu 17840 df-nat 17927 df-fuc 17928 df-setc 18059 df-xpc 18157 df-1stf 18158 df-2ndf 18159 df-prf 18160 df-curf 18200 df-hof 18236 df-yon 18237

This theorem is referenced by: yonedalem3a 18260 yonedalem3b 18265 yonedainv 18267 yonffthlem 18268

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