Theorem yonedalem3a 18344

Description: Lemma for yoneda 18353. (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 6924	. . . . . 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 7466	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
7	3	fveq2d 6924	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑎‘𝑥) = (𝑎‘𝑋))
8	3	fveq2d 6924	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
9	7, 8	fveq12d 6927	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑎‘𝑥)‘( 1 ‘𝑥)) = ((𝑎‘𝑋)‘( 1 ‘𝑋)))
10	6, 9	mpteq12dv 5257	. . . 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 7481	. . . . 5 ⊢ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
13	12	mptex 7260	. . . 4 ⊢ (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) ∈ V
14	10, 11, 13	ovmpoa 7605	. . 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 583	. 2 ⊢ (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))))
16		eqid 2740	. . . . . . 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 18019	. . . . . . 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 17777	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
22		eqid 2740	. . . . . . 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 18021	. . . . . 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 4217	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
29	26, 28	ssexd 5342	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ V)
30	29	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31		eqid 2740	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
32		relfunc 17926	. . . . . . . . . . . 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 18333	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37		1st2ndbr 8083	. . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑋)))
39	21, 31, 38	funcf1 17930	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
40	39, 2	ffvelcdmd 7119	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
41	25, 29	setcbas 18145	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
42	40, 41	eleqtrrd 2847	. . . . . . . 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 8083	. . . . . . . . . . . 12 ⊢ ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹))
45	32, 1, 44	sylancr 586	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹))
46	21, 31, 45	funcf1 17930	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝑆))
47	46, 2	ffvelcdmd 7119	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ (Base‘𝑆))
48	47, 41	eleqtrrd 2847	. . . . . . . 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 18148	. . . . . 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 2740	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53		yoneda.1	. . . . . . . 8 ⊢ 1 = (Id‘𝐶)
54	20, 52, 53, 34, 2	catidcl 17740	. . . . . . 7 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
55	33, 20, 34, 2, 52, 2	yon11 18334	. . . . . . 7 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
56	54, 55	eleqtrrd 2847	. . . . . 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 7119	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑋)‘( 1 ‘𝑋)) ∈ ((1st ‘𝐹)‘𝑋))
59	58	fmpttd 7149	. . 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 18343	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
67	19	oppccat 17782	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
68	34, 67	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Cat)
69	25	setccat 18152	. . . . . 6 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
70	29, 69	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Cat)
71	64, 68, 70, 21, 1, 2	evlf1 18290	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋))
72	15, 66, 71	feq123d 6736	. . 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 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  ⟨cop 4654   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701  Rel wrel 5705  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029  tpos ctpos 8266  Basecbs 17258  Hom chom 17322  Catccat 17722  Idccid 17723  Hom_f chomf 17724  oppCatcoppc 17769   Func cfunc 17918   ∘_func ccofu 17920   Nat cnat 18009   FuncCat cfuc 18010  SetCatcsetc 18142   ×_c cxpc 18237   1^st_F c1stf 18238   2^nd_F c2ndf 18239   ⟨,⟩_F cprf 18240   eval_F cevlf 18279  Hom_Fchof 18318  Yoncyon 18319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-hom 17335  df-cco 17336  df-cat 17726  df-cid 17727  df-homf 17728  df-comf 17729  df-oppc 17770  df-func 17922  df-cofu 17924  df-nat 18011  df-fuc 18012  df-setc 18143  df-xpc 18241  df-1stf 18242  df-2ndf 18243  df-prf 18244  df-evlf 18283  df-curf 18284  df-hof 18320  df-yon 18321

This theorem is referenced by:  yonedalem3b  18349  yonedalem3  18350  yonedainv  18351