Theorem yoniso 18212

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 17790	. . . 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 18029	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑉 ∩ Cat))
7		inss2 4191	. . . . . . 7 ⊢ (𝑉 ∩ Cat) ⊆ Cat
8	6, 7	eqsstrdi 3979	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ Cat)
9		yoniso.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
10	8, 9	sseldd 3935	. . . . 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 18189	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
17		1st2nd 7985	. . . 4 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
18	1, 16, 17	sylancr 588	. . 3 ⊢ (𝜑 → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
19	2, 11, 12, 13, 10, 14, 15	yonffth 18211	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
20	18, 19	eqeltrrd 2838	. . . 4 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21		eqid 2737	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		yoniso.e	. . . . . 6 ⊢ 𝐸 = (𝑄 ↾s ran (1st ‘𝑌))
23	11	oppccat 17649	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
24	10, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Cat)
25	12	setccat 18013	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑊 → 𝑆 ∈ Cat)
26	14, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Cat)
27	13, 24, 26	fuccat 17901	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Cat)
28		fvex 6848	. . . . . . . 8 ⊢ (1st ‘𝑌) ∈ V
29	28	rnex 7854	. . . . . . 7 ⊢ ran (1st ‘𝑌) ∈ V
30	29	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (1st ‘𝑌) ∈ V)
31	13	fucbas 17891	. . . . . . . . 9 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32		1st2ndbr 7988	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
33	1, 16, 32	sylancr 588	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
34	21, 31, 33	funcf1 17794	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
35	34	ffnd 6664	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌) Fn (Base‘𝐶))
36		dffn3 6675	. . . . . . 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 17870	. . . . 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 5100	. . . . 5 ⊢ ((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
40		df-br 5100	. . . . 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 2837	. 2 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
44		fveq2 6835	. . . . . . . . 9 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → (1st ‘((1st ‘𝑌)‘𝑥)) = (1st ‘((1st ‘𝑌)‘𝑦)))
45	44	fveq1d 6837	. . . . . . . 8 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → ((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥) = ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥))
46	45	fveq2d 6839	. . . . . . 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 2820	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
50	49	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
51	50	anbi2d 631	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
52		2fveq3 6840	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (1st ‘((1st ‘𝑌)‘𝑦)) = (1st ‘((1st ‘𝑌)‘𝑥)))
53	52	fveq1d 6837	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥) = ((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥))
54	53	fveq2d 6839	. . . . . . . . . . . 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 2753	. . . . . . . . . . 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 773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
60		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
61		simprl 771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
62	2, 21, 58, 59, 60, 61	yon11 18191	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
63	62	fveq2d 6839	. . . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦)
66	57, 65	chvarvv 1991	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
67	48, 66	sylan2 594	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
68	67, 65	eqeq12d 2753	. . . . . . 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 3180	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
71		dff13 7202	. . . . 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 584	. . . 4 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
73		f1f1orn 6786	. . . 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 6671	. . . . 5 ⊢ (𝜑 → ran (1st ‘𝑌) ⊆ (𝑂 Func 𝑆))
76	22, 31	ressbas2 17169	. . . . 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 6772	. . 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 2737	. . 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 18039	. 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 714	1 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  ⟨cop 4587   class class class wbr 5099  ran crn 5626  Rel wrel 5630   Fn wfn 6488  ⟶wf 6489  –1-1→wf1 6490  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17140   ↾_s cress 17161  Hom chom 17192  Catccat 17591  Hom_f chomf 17593  oppCatcoppc 17638  Isociso 17674   Func cfunc 17782   Full cful 17832   Faith cfth 17833   FuncCat cfuc 17873  SetCatcsetc 18003  CatCatccatc 18026  Yoncyon 18176

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12406  df-z 12493  df-dec 12612  df-uz 12756  df-fz 13428  df-struct 17078  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-hom 17205  df-cco 17206  df-cat 17595  df-cid 17596  df-homf 17597  df-comf 17598  df-oppc 17639  df-sect 17675  df-inv 17676  df-iso 17677  df-ssc 17738  df-resc 17739  df-subc 17740  df-func 17786  df-idfu 17787  df-cofu 17788  df-full 17834  df-fth 17835  df-nat 17874  df-fuc 17875  df-setc 18004  df-catc 18027  df-xpc 18099  df-1stf 18100  df-2ndf 18101  df-prf 18102  df-evlf 18140  df-curf 18141  df-hof 18177  df-yon 18178

This theorem is referenced by: (None)