Theorem yonedalem3a 17353

Description: Lemma for yoneda 17362. (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 485	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
4	3	fveq2d 6542	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑋))
5		simpl 483	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑓 = 𝐹)
6	4, 5	oveq12d 7034	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
7	3	fveq2d 6542	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑎‘𝑥) = (𝑎‘𝑋))
8	3	fveq2d 6542	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
9	7, 8	fveq12d 6545	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑎‘𝑥)‘( 1 ‘𝑥)) = ((𝑎‘𝑋)‘( 1 ‘𝑋)))
10	6, 9	mpteq12dv 5045	. . . 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 7048	. . . . 5 ⊢ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
13	12	mptex 6852	. . . 4 ⊢ (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) ∈ V
14	10, 11, 13	ovmpoa 7161	. . 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 2795	. . . . . . 7 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
17		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
18	16, 17	nat1st2nd 17050	. . . . . . 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 16817	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
22		eqid 2795	. . . . . . 7 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
23	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋 ∈ 𝐵)
24	16, 18, 21, 22, 23	natcl 17052	. . . . . 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 4085	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
29	26, 28	ssexd 5119	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ V)
30	29	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31		eqid 2795	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
32		relfunc 16961	. . . . . . . . . . . 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 17342	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37		1st2ndbr 7597	. . . . . . . . . . . 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 16965	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
40	39, 2	ffvelrnd 6717	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
41	25, 29	setcbas 17167	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
42	40, 41	eleqtrrd 2886	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
43	42	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
44		1st2ndbr 7597	. . . . . . . . . . . 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 16965	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝑆))
47	46, 2	ffvelrnd 6717	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ (Base‘𝑆))
48	47, 41	eleqtrrd 2886	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ 𝑈)
49	48	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘𝐹)‘𝑋) ∈ 𝑈)
50	25, 30, 22, 43, 49	elsetchom 17170	. . . . . 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 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎‘𝑋):((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋)⟶((1st ‘𝐹)‘𝑋))
52		eqid 2795	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53		yoneda.1	. . . . . . . 8 ⊢ 1 = (Id‘𝐶)
54	20, 52, 53, 34, 2	catidcl 16782	. . . . . . 7 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
55	33, 20, 34, 2, 52, 2	yon11 17343	. . . . . . 7 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
56	54, 55	eleqtrrd 2886	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋))
57	56	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1 ‘𝑋) ∈ ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋))
58	51, 57	ffvelrnd 6717	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑋)‘( 1 ‘𝑋)) ∈ ((1st ‘𝐹)‘𝑋))
59	58	fmpttd 6742	. . 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 17352	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
67	19	oppccat 16821	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
68	34, 67	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Cat)
69	25	setccat 17174	. . . . . 6 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
70	29, 69	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Cat)
71	64, 68, 70, 21, 1, 2	evlf1 17299	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋))
72	15, 66, 71	feq123d 6371	. . 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 258	. 2 ⊢ (𝜑 → (𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋))
74	15, 73	jca 512	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 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ∪ cun 3857   ⊆ wss 3859  ⟨cop 4478   class class class wbr 4962   ↦ cmpt 5041  ran crn 5444  Rel wrel 5448  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018  1^st c1st 7543  2^nd c2nd 7544  tpos ctpos 7742  Basecbs 16312  Hom chom 16405  Catccat 16764  Idccid 16765  Hom_f chomf 16766  oppCatcoppc 16810   Func cfunc 16953   ∘_func ccofu 16955   Nat cnat 17040   FuncCat cfuc 17041  SetCatcsetc 17164   ×_c cxpc 17247   1^st_F c1stf 17248   2^nd_F c2ndf 17249   ⟨,⟩_F cprf 17250   eval_F cevlf 17288  Hom_Fchof 17327  Yoncyon 17328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-tpos 7743  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-hom 16418  df-cco 16419  df-cat 16768  df-cid 16769  df-homf 16770  df-comf 16771  df-oppc 16811  df-func 16957  df-cofu 16959  df-nat 17042  df-fuc 17043  df-setc 17165  df-xpc 17251  df-1stf 17252  df-2ndf 17253  df-prf 17254  df-evlf 17292  df-curf 17293  df-hof 17329  df-yon 17330

This theorem is referenced by:  yonedalem3b  17358  yonedalem3  17359  yonedainv  17360