Theorem yonedalem3a 18320

Description: Lemma for yoneda 18329. (Contributed by Mario Carneiro, 29-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 = (HomF‘𝑄)
yoneda.r	⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e	⊢ 𝐸 = (𝑂 evalF 𝑆)
yoneda.z	⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yoneda.w	⊢ (𝜑 → 𝑉 ∈ 𝑊)
yoneda.u	⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
yoneda.v	⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
yonedalem21.f	⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
yonedalem3a.m	⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))))

Assertion

Ref	Expression
yonedalem3a	⊢ (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋)))

Distinct variable groups:   𝑓,𝑎,𝑥, 1   𝐶,𝑎,𝑓,𝑥   𝐸,𝑎,𝑓   𝐹,𝑎,𝑓,𝑥   𝐵,𝑎,𝑓,𝑥   𝑂,𝑎,𝑓,𝑥   𝑆,𝑎,𝑓,𝑥   𝑄,𝑎,𝑓,𝑥   𝑇,𝑓   𝜑,𝑎,𝑓,𝑥   𝑌,𝑎,𝑓,𝑥   𝑍,𝑎,𝑓,𝑥   𝑋,𝑎,𝑓,𝑥

Allowed substitution hints:   𝑅(𝑥,𝑓,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑓,𝑎)   𝐸(𝑥)   𝐻(𝑥,𝑓,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑉(𝑥,𝑓,𝑎)   𝑊(𝑥,𝑓,𝑎)

Proof of Theorem yonedalem3a

Step	Hyp	Ref	Expression
1		yonedalem21.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
2		yonedalem21.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		simpr 484	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
4	3	fveq2d 6909	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑋))
5		simpl 482	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑓 = 𝐹)
6	4, 5	oveq12d 7450	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
7	3	fveq2d 6909	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑎‘𝑥) = (𝑎‘𝑋))
8	3	fveq2d 6909	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
9	7, 8	fveq12d 6912	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑎‘𝑥)‘( 1 ‘𝑥)) = ((𝑎‘𝑋)‘( 1 ‘𝑋)))
10	6, 9	mpteq12dv 5232	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))))
11		yonedalem3a.m	. . . 4 ⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))))
12		ovex 7465	. . . . 5 ⊢ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
13	12	mptex 7244	. . . 4 ⊢ (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) ∈ V
14	10, 11, 13	ovmpoa 7589	. . 3 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))))
15	1, 2, 14	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))))
16		eqid 2736	. . . . . . 7 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
17		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
18	16, 17	nat1st2nd 18000	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (⟨(1st ‘((1st ‘𝑌)‘𝑋)), (2nd ‘((1st ‘𝑌)‘𝑋))⟩(𝑂 Nat 𝑆)⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
19		yoneda.o	. . . . . . . 8 ⊢ 𝑂 = (oppCat‘𝐶)
20		yoneda.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
21	19, 20	oppcbas 17762	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
22		eqid 2736	. . . . . . 7 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
23	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋 ∈ 𝐵)
24	16, 18, 21, 22, 23	natcl 18002	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎‘𝑋) ∈ (((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑋)))
25		yoneda.s	. . . . . . 7 ⊢ 𝑆 = (SetCat‘𝑈)
26		yoneda.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
27		yoneda.v	. . . . . . . . . 10 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
28	27	unssbd 4193	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
29	26, 28	ssexd 5323	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ V)
30	29	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
32		relfunc 17908	. . . . . . . . . . . 12 ⊢ Rel (𝑂 Func 𝑆)
33		yoneda.y	. . . . . . . . . . . . 13 ⊢ 𝑌 = (Yon‘𝐶)
34		yoneda.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ Cat)
35		yoneda.u	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
36	33, 20, 34, 2, 19, 25, 29, 35	yon1cl 18309	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37		1st2ndbr 8068	. . . . . . . . . . . 12 ⊢ ((Rel (𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑋)))
38	32, 36, 37	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑋)))
39	21, 31, 38	funcf1 17912	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
40	39, 2	ffvelcdmd 7104	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
41	25, 29	setcbas 18124	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
42	40, 41	eleqtrrd 2843	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
43	42	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
44		1st2ndbr 8068	. . . . . . . . . . . 12 ⊢ ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹))
45	32, 1, 44	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹))
46	21, 31, 45	funcf1 17912	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝑆))
47	46, 2	ffvelcdmd 7104	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ (Base‘𝑆))
48	47, 41	eleqtrrd 2843	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ 𝑈)
49	48	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘𝐹)‘𝑋) ∈ 𝑈)
50	25, 30, 22, 43, 49	elsetchom 18127	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑋) ∈ (((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋)(Hom ‘𝑆)((1st ‘𝐹)‘𝑋)) ↔ (𝑎‘𝑋):((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋)⟶((1st ‘𝐹)‘𝑋)))
51	24, 50	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎‘𝑋):((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋)⟶((1st ‘𝐹)‘𝑋))
52		eqid 2736	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53		yoneda.1	. . . . . . . 8 ⊢ 1 = (Id‘𝐶)
54	20, 52, 53, 34, 2	catidcl 17726	. . . . . . 7 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
55	33, 20, 34, 2, 52, 2	yon11 18310	. . . . . . 7 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
56	54, 55	eleqtrrd 2843	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋))
57	56	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1 ‘𝑋) ∈ ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋))
58	51, 57	ffvelcdmd 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑋)‘( 1 ‘𝑋)) ∈ ((1st ‘𝐹)‘𝑋))
59	58	fmpttd 7134	. . 3 ⊢ (𝜑 → (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))):(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st ‘𝐹)‘𝑋))
60		yoneda.t	. . . . 5 ⊢ 𝑇 = (SetCat‘𝑉)
61		yoneda.q	. . . . 5 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
62		yoneda.h	. . . . 5 ⊢ 𝐻 = (HomF‘𝑄)
63		yoneda.r	. . . . 5 ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
64		yoneda.e	. . . . 5 ⊢ 𝐸 = (𝑂 evalF 𝑆)
65		yoneda.z	. . . . 5 ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
66	33, 20, 53, 19, 25, 60, 61, 62, 63, 64, 65, 34, 26, 35, 27, 1, 2	yonedalem21 18319	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
67	19	oppccat 17766	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
68	34, 67	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Cat)
69	25	setccat 18131	. . . . . 6 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
70	29, 69	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Cat)
71	64, 68, 70, 21, 1, 2	evlf1 18266	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋))
72	15, 66, 71	feq123d 6724	. . 3 ⊢ (𝜑 → ((𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋) ↔ (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))):(((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)⟶((1st ‘𝐹)‘𝑋)))
73	59, 72	mpbird 257	. 2 ⊢ (𝜑 → (𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋))
74	15, 73	jca 511	1 ⊢ (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3479   ∪ cun 3948   ⊆ wss 3950  ⟨cop 4631   class class class wbr 5142   ↦ cmpt 5224  ran crn 5685  Rel wrel 5689  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434  1^st c1st 8013  2^nd c2nd 8014  tpos ctpos 8251  Basecbs 17248  Hom chom 17309  Catccat 17708  Idccid 17709  Hom_f chomf 17710  oppCatcoppc 17755   Func cfunc 17900   ∘_func ccofu 17902   Nat cnat 17990   FuncCat cfuc 17991  SetCatcsetc 18121   ×_c cxpc 18214   1^st_F c1stf 18215   2^nd_F c2ndf 18216   ⟨,⟩_F cprf 18217   eval_F cevlf 18255  Hom_Fchof 18294  Yoncyon 18295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  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 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-cat 17712  df-cid 17713  df-homf 17714  df-comf 17715  df-oppc 17756  df-func 17904  df-cofu 17906  df-nat 17992  df-fuc 17993  df-setc 18122  df-xpc 18218  df-1stf 18219  df-2ndf 18220  df-prf 18221  df-evlf 18259  df-curf 18260  df-hof 18296  df-yon 18297

This theorem is referenced by:  yonedalem3b  18325  yonedalem3  18326  yonedainv  18327