Theorem yonedalem3a 18163

Description: Lemma for yoneda 18172. (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 6846	. . . . . 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 7375	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
7	3	fveq2d 6846	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑎‘𝑥) = (𝑎‘𝑋))
8	3	fveq2d 6846	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
9	7, 8	fveq12d 6849	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑎‘𝑥)‘( 1 ‘𝑥)) = ((𝑎‘𝑋)‘( 1 ‘𝑋)))
10	6, 9	mpteq12dv 5196	. . . 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 7390	. . . . 5 ⊢ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
13	12	mptex 7173	. . . 4 ⊢ (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) ∈ V
14	10, 11, 13	ovmpoa 7510	. . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
18	16, 17	nat1st2nd 17838	. . . . . . 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 17599	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
22		eqid 2736	. . . . . . 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 17840	. . . . . 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 4148	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
29	26, 28	ssexd 5281	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ V)
30	29	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
32		relfunc 17748	. . . . . . . . . . . 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 18152	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37		1st2ndbr 7974	. . . . . . . . . . . 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 17752	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
40	39, 2	ffvelcdmd 7036	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
41	25, 29	setcbas 17964	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
42	40, 41	eleqtrrd 2841	. . . . . . . 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 7974	. . . . . . . . . . . 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 17752	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝑆))
47	46, 2	ffvelcdmd 7036	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ (Base‘𝑆))
48	47, 41	eleqtrrd 2841	. . . . . . . 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 17967	. . . . . 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 231	. . . . 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 17562	. . . . . . 7 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
55	33, 20, 34, 2, 52, 2	yon11 18153	. . . . . . 7 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
56	54, 55	eleqtrrd 2841	. . . . . 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	ffvelcdmd 7036	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑋)‘( 1 ‘𝑋)) ∈ ((1st ‘𝐹)‘𝑋))
59	58	fmpttd 7063	. . 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 18162	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
67	19	oppccat 17604	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
68	34, 67	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Cat)
69	25	setccat 17971	. . . . . 6 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
70	29, 69	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Cat)
71	64, 68, 70, 21, 1, 2	evlf1 18109	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋))
72	15, 66, 71	feq123d 6657	. . 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 256	. 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 1541   ∈ wcel 2106  Vcvv 3445   ∪ cun 3908   ⊆ wss 3910  ⟨cop 4592   class class class wbr 5105   ↦ cmpt 5188  ran crn 5634  Rel wrel 5638  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  1^st c1st 7919  2^nd c2nd 7920  tpos ctpos 8156  Basecbs 17083  Hom chom 17144  Catccat 17544  Idccid 17545  Hom_f chomf 17546  oppCatcoppc 17591   Func cfunc 17740   ∘_func ccofu 17742   Nat cnat 17828   FuncCat cfuc 17829  SetCatcsetc 17961   ×_c cxpc 18056   1^st_F c1stf 18057   2^nd_F c2ndf 18058   ⟨,⟩_F cprf 18059   eval_F cevlf 18098  Hom_Fchof 18137  Yoncyon 18138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-homf 17550  df-comf 17551  df-oppc 17592  df-func 17744  df-cofu 17746  df-nat 17830  df-fuc 17831  df-setc 17962  df-xpc 18060  df-1stf 18061  df-2ndf 18062  df-prf 18063  df-evlf 18102  df-curf 18103  df-hof 18139  df-yon 18140

This theorem is referenced by:  yonedalem3b  18168  yonedalem3  18169  yonedainv  18170