Theorem yoniso 18375

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 17953	. . . 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 18192	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑉 ∩ Cat))
7		inss2 4186	. . . . . . 7 ⊢ (𝑉 ∩ Cat) ⊆ Cat
8	6, 7	eqsstrdi 3978	. . . . . 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 18352	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
17		1st2nd 8039	. . . 4 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
18	1, 16, 17	sylancr 599	. . 3 ⊢ (𝜑 → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
19	2, 11, 12, 13, 10, 14, 15	yonffth 18374	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
20	18, 19	eqeltrrd 2863	. . . 4 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21		eqid 2762	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		yoniso.e	. . . . . 6 ⊢ 𝐸 = (𝑄 ↾s ran (1st ‘𝑌))
23	11	oppccat 17812	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
24	10, 23	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Cat)
25	12	setccat 18176	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑊 → 𝑆 ∈ Cat)
26	14, 25	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Cat)
27	13, 24, 26	fuccat 18064	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Cat)
28		fvex 6895	. . . . . . . 8 ⊢ (1st ‘𝑌) ∈ V
29	28	rnex 7910	. . . . . . 7 ⊢ ran (1st ‘𝑌) ∈ V
30	29	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (1st ‘𝑌) ∈ V)
31	13	fucbas 18054	. . . . . . . . 9 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32		1st2ndbr 8042	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
33	1, 16, 32	sylancr 599	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
34	21, 31, 33	funcf1 17957	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
35	34	ffnd 6707	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌) Fn (Base‘𝐶))
36		dffn3 6719	. . . . . . 7 ⊢ ((1st ‘𝑌) Fn (Base‘𝐶) ↔ (1st ‘𝑌):(Base‘𝐶)⟶ran (1st ‘𝑌))
37	35, 36	sylib 221	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)⟶ran (1st ‘𝑌))
38	21, 22, 27, 30, 37	ffthres2c 18033	. . . . 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 5108	. . . . 5 ⊢ ((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
40		df-br 5108	. . . . 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 316	. . . 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 235	. . 3 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
43	18, 42	eqeltrd 2862	. 2 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
44		fveq2 6882	. . . . . . . . 9 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → (1st ‘((1st ‘𝑌)‘𝑥)) = (1st ‘((1st ‘𝑌)‘𝑦)))
45	44	fveq1d 6884	. . . . . . . 8 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → ((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥) = ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥))
46	45	fveq2d 6886	. . . . . . 7 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)))
47		simpl 488	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
48	47, 47	jca 521	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
49		eleq1w 2845	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
50	49	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
51	50	anbi2d 642	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
52		2fveq3 6887	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (1st ‘((1st ‘𝑌)‘𝑦)) = (1st ‘((1st ‘𝑌)‘𝑥)))
53	52	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥) = ((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥))
54	53	fveq2d 6886	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)))
55		id 23	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
56	54, 55	eqeq12d 2778	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦 ↔ (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥))
57	51, 56	imbi12d 347	. . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
59		simprr 785	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
60		eqid 2762	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
61		simprl 783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
62	2, 21, 58, 59, 60, 61	yon11 18354	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
63	62	fveq2d 6886	. . . . . . . . . . 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 2797	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦)
66	57, 65	chvarvv 2022	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
67	48, 66	sylan2 605	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
68	67, 65	eqeq12d 2778	. . . . . . 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 247	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
70	69	ralrimivva 3207	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
71		dff13 7254	. . . . 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 595	. . . 4 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
73		f1f1orn 6833	. . . 4 ⊢ ((1st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆) → (1st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1st ‘𝑌))
74	72, 73	syl 18	. . 3 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1st ‘𝑌))
75	34	frnd 6715	. . . . 5 ⊢ (𝜑 → ran (1st ‘𝑌) ⊆ (𝑂 Func 𝑆))
76	22, 31	ressbas2 17332	. . . . 5 ⊢ (ran (1st ‘𝑌) ⊆ (𝑂 Func 𝑆) → ran (1st ‘𝑌) = (Base‘𝐸))
77	75, 76	syl 18	. . . 4 ⊢ (𝜑 → ran (1st ‘𝑌) = (Base‘𝐸))
78	77	f1oeq3d 6818	. . 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 235	. 2 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))
80		eqid 2762	. . 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 18202	. 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 726	1 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ⟨cop 4593   class class class wbr 5107  ran crn 5660  Rel wrel 5664   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416  1^st c1st 7987  2^nd c2nd 7988  Basecbs 17303   ↾_s cress 17324  Hom chom 17355  Catccat 17754  Hom_f chomf 17756  oppCatcoppc 17801  Isociso 17837   Func cfunc 17945   Full cful 17995   Faith cfth 17996   FuncCat cfuc 18036  SetCatcsetc 18166  CatCatccatc 18189  Yoncyon 18339

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-tpos 8227  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-pm 8832  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-9 12335  df-n0 12530  df-z 12617  df-dec 12738  df-uz 12889  df-fz 13562  df-struct 17241  df-sets 17258  df-slot 17276  df-ndx 17288  df-base 17304  df-ress 17325  df-hom 17368  df-cco 17369  df-cat 17758  df-cid 17759  df-homf 17760  df-comf 17761  df-oppc 17802  df-sect 17838  df-inv 17839  df-iso 17840  df-ssc 17901  df-resc 17902  df-subc 17903  df-func 17949  df-idfu 17950  df-cofu 17951  df-full 17997  df-fth 17998  df-nat 18037  df-fuc 18038  df-setc 18167  df-catc 18190  df-xpc 18262  df-1stf 18263  df-2ndf 18264  df-prf 18265  df-evlf 18303  df-curf 18304  df-hof 18340  df-yon 18341

This theorem is used by: (None)