Theorem yoniso 18206

Description: If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from 𝐶 into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)

Hypotheses

Ref	Expression
yoniso.y	⊢ 𝑌 = (Yon‘𝐶)
yoniso.o	⊢ 𝑂 = (oppCat‘𝐶)
yoniso.s	⊢ 𝑆 = (SetCat‘𝑈)
yoniso.d	⊢ 𝐷 = (CatCat‘𝑉)
yoniso.b	⊢ 𝐵 = (Base‘𝐷)
yoniso.i	⊢ 𝐼 = (Iso‘𝐷)
yoniso.q	⊢ 𝑄 = (𝑂 FuncCat 𝑆)
yoniso.e	⊢ 𝐸 = (𝑄 ↾s ran (1st ‘𝑌))
yoniso.v	⊢ (𝜑 → 𝑉 ∈ 𝑋)
yoniso.c	⊢ (𝜑 → 𝐶 ∈ 𝐵)
yoniso.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
yoniso.h	⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
yoniso.eb	⊢ (𝜑 → 𝐸 ∈ 𝐵)
yoniso.1	⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘(𝑥(Hom ‘𝐶)𝑦)) = 𝑦)

Assertion

Ref	Expression
yoniso	⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝐸))

Distinct variable groups:   𝑥,𝑦,𝐶   𝑦,𝐹   𝜑,𝑥,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑄(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥)   𝐼(𝑥,𝑦)   𝑂(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem yoniso

Step	Hyp	Ref	Expression
1		relfunc 17784	. . . 4 ⊢ Rel (𝐶 Func 𝑄)
2		yoniso.y	. . . . 5 ⊢ 𝑌 = (Yon‘𝐶)
3		yoniso.d	. . . . . . . 8 ⊢ 𝐷 = (CatCat‘𝑉)
4		yoniso.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐷)
5		yoniso.v	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑋)
6	3, 4, 5	catcbas 18023	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑉 ∩ Cat))
7		inss2 4188	. . . . . . 7 ⊢ (𝑉 ∩ Cat) ⊆ Cat
8	6, 7	eqsstrdi 3976	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ Cat)
9		yoniso.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
10	8, 9	sseldd 3932	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		yoniso.o	. . . . 5 ⊢ 𝑂 = (oppCat‘𝐶)
12		yoniso.s	. . . . 5 ⊢ 𝑆 = (SetCat‘𝑈)
13		yoniso.q	. . . . 5 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
14		yoniso.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
15		yoniso.h	. . . . 5 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
16	2, 10, 11, 12, 13, 14, 15	yoncl 18183	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
17		1st2nd 7981	. . . 4 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
18	1, 16, 17	sylancr 587	. . 3 ⊢ (𝜑 → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
19	2, 11, 12, 13, 10, 14, 15	yonffth 18205	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
20	18, 19	eqeltrrd 2835	. . . 4 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21		eqid 2734	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		yoniso.e	. . . . . 6 ⊢ 𝐸 = (𝑄 ↾s ran (1st ‘𝑌))
23	11	oppccat 17643	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
24	10, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Cat)
25	12	setccat 18007	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑊 → 𝑆 ∈ Cat)
26	14, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Cat)
27	13, 24, 26	fuccat 17895	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Cat)
28		fvex 6845	. . . . . . . 8 ⊢ (1st ‘𝑌) ∈ V
29	28	rnex 7850	. . . . . . 7 ⊢ ran (1st ‘𝑌) ∈ V
30	29	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (1st ‘𝑌) ∈ V)
31	13	fucbas 17885	. . . . . . . . 9 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32		1st2ndbr 7984	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
33	1, 16, 32	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
34	21, 31, 33	funcf1 17788	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
35	34	ffnd 6661	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌) Fn (Base‘𝐶))
36		dffn3 6672	. . . . . . 7 ⊢ ((1st ‘𝑌) Fn (Base‘𝐶) ↔ (1st ‘𝑌):(Base‘𝐶)⟶ran (1st ‘𝑌))
37	35, 36	sylib 218	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)⟶ran (1st ‘𝑌))
38	21, 22, 27, 30, 37	ffthres2c 17864	. . . . 5 ⊢ (𝜑 → ((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ (1st ‘𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd ‘𝑌)))
39		df-br 5097	. . . . 5 ⊢ ((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
40		df-br 5097	. . . . 5 ⊢ ((1st ‘𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
41	38, 39, 40	3bitr3g 313	. . . 4 ⊢ (𝜑 → (⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))))
42	20, 41	mpbid 232	. . 3 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
43	18, 42	eqeltrd 2834	. 2 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
44		fveq2 6832	. . . . . . . . 9 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → (1st ‘((1st ‘𝑌)‘𝑥)) = (1st ‘((1st ‘𝑌)‘𝑦)))
45	44	fveq1d 6834	. . . . . . . 8 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → ((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥) = ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥))
46	45	fveq2d 6836	. . . . . . 7 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)))
47		simpl 482	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
48	47, 47	jca 511	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
49		eleq1w 2817	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
50	49	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
51	50	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
52		2fveq3 6837	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (1st ‘((1st ‘𝑌)‘𝑦)) = (1st ‘((1st ‘𝑌)‘𝑥)))
53	52	fveq1d 6834	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥) = ((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥))
54	53	fveq2d 6836	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)))
55		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
56	54, 55	eqeq12d 2750	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦 ↔ (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥))
57	51, 56	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦) ↔ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)))
58	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
59		simprr 772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
60		eqid 2734	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
61		simprl 770	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
62	2, 21, 58, 59, 60, 61	yon11 18185	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
63	62	fveq2d 6836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = (𝐹‘(𝑥(Hom ‘𝐶)𝑦)))
64		yoniso.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘(𝑥(Hom ‘𝐶)𝑦)) = 𝑦)
65	63, 64	eqtrd 2769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦)
66	57, 65	chvarvv 1990	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
67	48, 66	sylan2 593	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
68	67, 65	eqeq12d 2750	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) ↔ 𝑥 = 𝑦))
69	46, 68	imbitrid 244	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
70	69	ralrimivva 3177	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
71		dff13 7198	. . . . 5 ⊢ ((1st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆) ↔ ((1st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → 𝑥 = 𝑦)))
72	34, 70, 71	sylanbrc 583	. . . 4 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
73		f1f1orn 6783	. . . 4 ⊢ ((1st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆) → (1st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1st ‘𝑌))
74	72, 73	syl 17	. . 3 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1st ‘𝑌))
75	34	frnd 6668	. . . . 5 ⊢ (𝜑 → ran (1st ‘𝑌) ⊆ (𝑂 Func 𝑆))
76	22, 31	ressbas2 17163	. . . . 5 ⊢ (ran (1st ‘𝑌) ⊆ (𝑂 Func 𝑆) → ran (1st ‘𝑌) = (Base‘𝐸))
77	75, 76	syl 17	. . . 4 ⊢ (𝜑 → ran (1st ‘𝑌) = (Base‘𝐸))
78	77	f1oeq3d 6769	. . 3 ⊢ (𝜑 → ((1st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1st ‘𝑌) ↔ (1st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸)))
79	74, 78	mpbid 232	. 2 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))
80		eqid 2734	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
81		yoniso.eb	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
82		yoniso.i	. . 3 ⊢ 𝐼 = (Iso‘𝐷)
83	3, 4, 21, 80, 5, 9, 81, 82	catciso 18033	. 2 ⊢ (𝜑 → (𝑌 ∈ (𝐶𝐼𝐸) ↔ (𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)) ∧ (1st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))))
84	43, 79, 83	mpbir2and 713	1 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  Vcvv 3438   ∩ cin 3898   ⊆ wss 3899  ⟨cop 4584   class class class wbr 5096  ran crn 5623  Rel wrel 5627   Fn wfn 6485  ⟶wf 6486  –1-1→wf1 6487  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17134   ↾_s cress 17155  Hom chom 17186  Catccat 17585  Hom_f chomf 17587  oppCatcoppc 17632  Isociso 17668   Func cfunc 17776   Full cful 17826   Faith cfth 17827   FuncCat cfuc 17867  SetCatcsetc 17997  CatCatccatc 18020  Yoncyon 18170

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-pm 8764  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-hom 17199  df-cco 17200  df-cat 17589  df-cid 17590  df-homf 17591  df-comf 17592  df-oppc 17633  df-sect 17669  df-inv 17670  df-iso 17671  df-ssc 17732  df-resc 17733  df-subc 17734  df-func 17780  df-idfu 17781  df-cofu 17782  df-full 17828  df-fth 17829  df-nat 17868  df-fuc 17869  df-setc 17998  df-catc 18021  df-xpc 18093  df-1stf 18094  df-2ndf 18095  df-prf 18096  df-evlf 18134  df-curf 18135  df-hof 18171  df-yon 18172

This theorem is referenced by: (None)