Theorem yonedalem3a 17524

Description: Lemma for yoneda 17533. (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 487	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
4	3	fveq2d 6674	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑋))
5		simpl 485	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑓 = 𝐹)
6	4, 5	oveq12d 7174	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
7	3	fveq2d 6674	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑎‘𝑥) = (𝑎‘𝑋))
8	3	fveq2d 6674	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
9	7, 8	fveq12d 6677	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑎‘𝑥)‘( 1 ‘𝑥)) = ((𝑎‘𝑋)‘( 1 ‘𝑋)))
10	6, 9	mpteq12dv 5151	. . . 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 7189	. . . . 5 ⊢ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ∈ V
13	12	mptex 6986	. . . 4 ⊢ (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) ∈ V
14	10, 11, 13	ovmpoa 7305	. . 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 586	. 2 ⊢ (𝜑 → (𝐹𝑀𝑋) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))))
16		eqid 2821	. . . . . . 7 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
17		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
18	16, 17	nat1st2nd 17221	. . . . . . 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 16988	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
22		eqid 2821	. . . . . . 7 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
23	2	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑋 ∈ 𝐵)
24	16, 18, 21, 22, 23	natcl 17223	. . . . . 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 4164	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
29	26, 28	ssexd 5228	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ V)
30	29	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → 𝑈 ∈ V)
31		eqid 2821	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
32		relfunc 17132	. . . . . . . . . . . 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 17513	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
37		1st2ndbr 7741	. . . . . . . . . . . 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 589	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑋)))
39	21, 31, 38	funcf1 17136	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
40	39, 2	ffvelrnd 6852	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ (Base‘𝑆))
41	25, 29	setcbas 17338	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
42	40, 41	eleqtrrd 2916	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
43	42	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) ∈ 𝑈)
44		1st2ndbr 7741	. . . . . . . . . . . 12 ⊢ ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹))
45	32, 1, 44	sylancr 589	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘𝐹)(𝑂 Func 𝑆)(2nd ‘𝐹))
46	21, 31, 45	funcf1 17136	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝑆))
47	46, 2	ffvelrnd 6852	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ (Base‘𝑆))
48	47, 41	eleqtrrd 2916	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ 𝑈)
49	48	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((1st ‘𝐹)‘𝑋) ∈ 𝑈)
50	25, 30, 22, 43, 49	elsetchom 17341	. . . . . 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 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → (𝑎‘𝑋):((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋)⟶((1st ‘𝐹)‘𝑋))
52		eqid 2821	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53		yoneda.1	. . . . . . . 8 ⊢ 1 = (Id‘𝐶)
54	20, 52, 53, 34, 2	catidcl 16953	. . . . . . 7 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
55	33, 20, 34, 2, 52, 2	yon11 17514	. . . . . . 7 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋) = (𝑋(Hom ‘𝐶)𝑋))
56	54, 55	eleqtrrd 2916	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋))
57	56	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ( 1 ‘𝑋) ∈ ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑋))
58	51, 57	ffvelrnd 6852	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹)) → ((𝑎‘𝑋)‘( 1 ‘𝑋)) ∈ ((1st ‘𝐹)‘𝑋))
59	58	fmpttd 6879	. . 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 17523	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
67	19	oppccat 16992	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
68	34, 67	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Cat)
69	25	setccat 17345	. . . . . 6 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
70	29, 69	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Cat)
71	64, 68, 70, 21, 1, 2	evlf1 17470	. . . 4 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋))
72	15, 66, 71	feq123d 6503	. . 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 259	. 2 ⊢ (𝜑 → (𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋))
74	15, 73	jca 514	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 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∪ cun 3934   ⊆ wss 3936  ⟨cop 4573   class class class wbr 5066   ↦ cmpt 5146  ran crn 5556  Rel wrel 5560  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  1^st c1st 7687  2^nd c2nd 7688  tpos ctpos 7891  Basecbs 16483  Hom chom 16576  Catccat 16935  Idccid 16936  Hom_f chomf 16937  oppCatcoppc 16981   Func cfunc 17124   ∘_func ccofu 17126   Nat cnat 17211   FuncCat cfuc 17212  SetCatcsetc 17335   ×_c cxpc 17418   1^st_F c1stf 17419   2^nd_F c2ndf 17420   ⟨,⟩_F cprf 17421   eval_F cevlf 17459  Hom_Fchof 17498  Yoncyon 17499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-tpos 7892  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-fz 12894  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-hom 16589  df-cco 16590  df-cat 16939  df-cid 16940  df-homf 16941  df-comf 16942  df-oppc 16982  df-func 17128  df-cofu 17130  df-nat 17213  df-fuc 17214  df-setc 17336  df-xpc 17422  df-1stf 17423  df-2ndf 17424  df-prf 17425  df-evlf 17463  df-curf 17464  df-hof 17500  df-yon 17501

This theorem is referenced by:  yonedalem3b  17529  yonedalem3  17530  yonedainv  17531