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Theorem yonedalem21 18365

Description: Lemma for yoneda 18375. (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 6885	. . . . 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 7429	. . . 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 7419	. . . 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 2785	. . 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 2762	. . . . 5 ⊢ (𝑄 ×_c 𝑂) = (𝑄 ×_c 𝑂)
7		yoneda.q	. . . . . 6 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8	7	fucbas 18056	. . . . 5 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
9		yoneda.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
10		yoneda.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	oppcbas 17810	. . . . 5 ⊢ 𝐵 = (Base‘𝑂)
12	6, 8, 11	xpcbas 18270	. . . 4 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×_c 𝑂))
13		eqid 2762	. . . . 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 2762	. . . . 5 ⊢ ((oppCat‘𝑄) ×_c 𝑄) = ((oppCat‘𝑄) ×_c 𝑄)
15		yoneda.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	9	oppccat 17814	. . . . . . . . 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 4143	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
21	18, 20	ssexd 5293	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
22		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
23	22	setccat 18178	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
24	21, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
25	7, 17, 24	fuccat 18066	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
26		eqid 2762	. . . . . . 7 ⊢ (𝑄 2^nd_F 𝑂) = (𝑄 2^nd_F 𝑂)
27	6, 25, 17, 26	2ndfcl 18290	. . . . . 6 ⊢ (𝜑 → (𝑄 2^nd_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑂))
28		eqid 2762	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
29		relfunc 17955	. . . . . . . . 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 18354	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
33		1st2ndbr 8042	. . . . . . . . 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 17968	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2^nd ‘𝑌))
36		df-br 5108	. . . . . . 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 17981	. . . . 5 ⊢ (𝜑 → (⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ∈ ((𝑄 ×_c 𝑂) Func (oppCat‘𝑄)))
39		eqid 2762	. . . . . 6 ⊢ (𝑄 1^st_F 𝑂) = (𝑄 1^st_F 𝑂)
40	6, 25, 17, 39	1stfcl 18289	. . . . 5 ⊢ (𝜑 → (𝑄 1^st_F 𝑂) ∈ ((𝑄 ×_c 𝑂) Func 𝑄))
41	13, 14, 38, 40	prfcl 18295	. . . 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 4142	. . . . 5 ⊢ (𝜑 → ran (Hom_f ‘𝑄) ⊆ 𝑉)
45	42, 28, 43, 25, 18, 44	hofcl 18351	. . . 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 5698	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
49	12, 41, 45, 48	cofu1 17977	. . 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 2809	. 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 2762	. . . . . 6 ⊢ (Hom ‘(𝑄 ×_c 𝑂)) = (Hom ‘(𝑄 ×_c 𝑂))
52	13, 12, 51, 38, 40, 48	prf1 18292	. . . . 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 17977	. . . . . . 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 6895	. . . . . . . . . 10 ⊢ (1^st ‘𝑌) ∈ V
55		fvex 6895	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑌) ∈ V
56	55	tposex 8261	. . . . . . . . . 10 ⊢ tpos (2^nd ‘𝑌) ∈ V
57	54, 56	op1st 7997	. . . . . . . . 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 18287	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = (2^nd ‘⟨𝐹, 𝑋⟩))
60		op2ndg 8002	. . . . . . . . . 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 2797	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘(𝑄 2^nd_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
63	58, 62	fveq12d 6889	. . . . . . 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 2797	. . . . . 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 18284	. . . . . . 7 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = (1^st ‘⟨𝐹, 𝑋⟩))
66		op1stg 8001	. . . . . . . 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 2797	. . . . . 6 ⊢ (𝜑 → ((1^st ‘(𝑄 1^st_F 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
69	64, 68	opeq12d 4844	. . . . 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 2797	. . . 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 6886	. . 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 7419	. . 3 ⊢ (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = ((1^st ‘𝐻)‘⟨((1^st ‘𝑌)‘𝑋), 𝐹⟩)
73	71, 72	eqtr4di 2815	. 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 2762	. . . 4 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
75	7, 74	fuchom 18057	. . 3 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
76	30, 10, 15, 47, 9, 22, 21, 31	yon1cl 18355	. . 3 ⊢ (𝜑 → ((1^st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
77	42, 25, 8, 75, 76, 46	hof1 18346	. 2 ⊢ (𝜑 → (((1^st ‘𝑌)‘𝑋)(1^st ‘𝐻)𝐹) = (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
78	50, 73, 77	3eqtrd 2801	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 3453 ∪ cun 3900 ⊆ wss 3902 ⟨cop 4593 class class class wbr 5107 × cxp 5657 ran crn 5660 Rel wrel 5664 ‘cfv 6537 (class class class)co 7416 1^st c1st 7987 2^nd c2nd 7988 tpos ctpos 8226 Basecbs 17305 Hom chom 17357 Catccat 17756 Idccid 17757 Hom_f chomf 17758 oppCatcoppc 17803 Func cfunc 17947 ∘_func ccofu 17949 Nat cnat 18037 FuncCat cfuc 18038 SetCatcsetc 18168 ×_c cxpc 18260 1^st_F c1stf 18261 2^nd_F c2ndf 18262 ⟨,⟩_F cprf 18263 eval_F cevlf 18301 Hom_Fchof 18340 Yoncyon 18341

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-fz 13564 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-hom 17370 df-cco 17371 df-cat 17760 df-cid 17761 df-homf 17762 df-comf 17763 df-oppc 17804 df-func 17951 df-cofu 17953 df-nat 18039 df-fuc 18040 df-setc 18169 df-xpc 18264 df-1stf 18265 df-2ndf 18266 df-prf 18267 df-curf 18306 df-hof 18342 df-yon 18343

This theorem is used by: yonedalem3a 18366 yonedalem3b 18371 yonedainv 18373 yonffthlem 18374

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