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Theorem yonedalem21 18408

Description: Lemma for yoneda 18418. (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 6876	. . . . 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 7421	. . . 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 7411	. . . 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 2783	. . 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 2760	. . . . 5 ⊢ (𝑄 ×_c 𝑂) = (𝑄 ×_c 𝑂)
7		yoneda.q	. . . . . 6 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8	7	fucbas 18099	. . . . 5 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
9		yoneda.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
10		yoneda.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	oppcbas 17853	. . . . 5 ⊢ 𝐵 = (Base‘𝑂)
12	6, 8, 11	xpcbas 18313	. . . 4 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×_c 𝑂))
13		eqid 2760	. . . . 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 2760	. . . . 5 ⊢ ((oppCat‘𝑄) ×_c 𝑄) = ((oppCat‘𝑄) ×_c 𝑄)
15		yoneda.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	9	oppccat 17857	. . . . . . . . 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 4139	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
21	18, 20	ssexd 5285	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
22		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
23	22	setccat 18221	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
24	21, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
25	7, 17, 24	fuccat 18109	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
26		eqid 2760	. . . . . . 7 ⊢ (𝑄 2^nd_F 𝑂) = (𝑄 2^nd_F 𝑂)
27	6, 25, 17, 26	2ndfcl 18333	. . . . . 6 ⊢ (𝜑 → (𝑄 2^nd_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑂))
28		eqid 2760	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
29		relfunc 17998	. . . . . . . . 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 18397	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
33		1st2ndbr 8036	. . . . . . . . 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 18011	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2^nd ‘𝑌))
36		df-br 5103	. . . . . . 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 18024	. . . . 5 ⊢ (𝜑 → (⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ∈ ((𝑄 ×_c 𝑂) Func (oppCat‘𝑄)))
39		eqid 2760	. . . . . 6 ⊢ (𝑄 1^st_F 𝑂) = (𝑄 1^st_F 𝑂)
40	6, 25, 17, 39	1stfcl 18332	. . . . 5 ⊢ (𝜑 → (𝑄 1^st_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑄))
41	13, 14, 38, 40	prfcl 18338	. . . 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 4138	. . . . 5 ⊢ (𝜑 → ran (Hom_f ‘𝑄) ⊆ 𝑉)
45	42, 28, 43, 25, 18, 44	hofcl 18394	. . . 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 5686	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
49	12, 41, 45, 48	cofu1 18020	. . 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 2807	. 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 2760	. . . . . 6 ⊢ (Hom ‘(𝑄 ×_c 𝑂)) = (Hom ‘(𝑄 ×_c 𝑂))
52	13, 12, 51, 38, 40, 48	prf1 18335	. . . . 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 18020	. . . . . . 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 6886	. . . . . . . . . 10 ⊢ (1^st ‘𝑌) ∈ V
55		fvex 6886	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑌) ∈ V
56	55	tposex 8255	. . . . . . . . . 10 ⊢ tpos (2^nd ‘𝑌) ∈ V
57	54, 56	op1st 7992	. . . . . . . . 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 18330	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = (2^nd ‘⟨𝐹, 𝑋⟩))
60		op2ndg 7997	. . . . . . . . . 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 2795	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
63	58, 62	fveq12d 6880	. . . . . . 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 2795	. . . . . 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 18327	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = (1^st ‘⟨𝐹, 𝑋⟩))
66		op1stg 7996	. . . . . . . 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 2795	. . . . . 6 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
69	64, 68	opeq12d 4840	. . . . 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 2795	. . . 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 6877	. . 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 7411	. . 3 ⊢ (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = ((1^st ‘𝐻)‘⟨((1^st ‘𝑌)‘𝑋), 𝐹⟩)
73	71, 72	eqtr4di 2813	. 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 2760	. . . 4 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
75	7, 74	fuchom 18100	. . 3 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
76	30, 10, 15, 47, 9, 22, 21, 31	yon1cl 18398	. . 3 ⊢ (𝜑 → ((1^st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
77	42, 25, 8, 75, 76, 46	hof1 18389	. 2 ⊢ (𝜑 → (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
78	50, 73, 77	3eqtrd 2799	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 2145 Vcvv 3450 ∪ cun 3896 ⊆ wss 3898 ⟨cop 4589 class class class wbr 5102 × cxp 5645 ran crn 5648 Rel wrel 5652 ‘cfv 6527 (class class class)co 7408 1^st c1st 7982 2^nd c2nd 7983 tpos ctpos 8220 Basecbs 17348 Hom chom 17400 Catccat 17799 Idccid 17800 Hom_f chomf 17801 oppCatcoppc 17846 Func cfunc 17990 ∘_func ccofu 17992 Nat cnat 18080 FuncCat cfuc 18081 SetCatcsetc 18211 ×_c cxpc 18303 1^st_F c1stf 18304 2^nd_F c2ndf 18305 ⟨,⟩_F cprf 18306 eval_F cevlf 18344 Hom_Fchof 18383 Yoncyon 18384

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-er 8695 df-map 8827 df-ixp 8904 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-nn 12305 df-2 12374 df-3 12375 df-4 12376 df-5 12377 df-6 12378 df-7 12379 df-8 12380 df-9 12381 df-n0 12576 df-z 12663 df-dec 12784 df-uz 12935 df-fz 13609 df-struct 17286 df-sets 17303 df-slot 17321 df-ndx 17333 df-base 17349 df-hom 17413 df-cco 17414 df-cat 17803 df-cid 17804 df-homf 17805 df-comf 17806 df-oppc 17847 df-func 17994 df-cofu 17996 df-nat 18082 df-fuc 18083 df-setc 18212 df-xpc 18307 df-1stf 18308 df-2ndf 18309 df-prf 18310 df-curf 18349 df-hof 18385 df-yon 18386

This theorem is used by: yonedalem3a 18409 yonedalem3b 18414 yonedainv 18416 yonffthlem 18417

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