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Theorem yoniso 18398

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 (1^st ‘𝑌))
yoniso.v	⊢ (𝜑 → 𝑉 ∈ 𝑋)
yoniso.c	⊢ (𝜑 → 𝐶 ∈ 𝐵)
yoniso.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
yoniso.h	⊢ (𝜑 → ran (Hom_f ‘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 17976	. . . 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 18215	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑉 ∩ Cat))
7		inss2 4183	. . . . . . 7 ⊢ (𝑉 ∩ Cat) ⊆ Cat
8	6, 7	eqsstrdi 3975	. . . . . 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 (Hom_f ‘𝐶) ⊆ 𝑈)
16	2, 10, 11, 12, 13, 14, 15	yoncl 18375	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
17		1st2nd 8041	. . . 4 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
18	1, 16, 17	sylancr 599	. . 3 ⊢ (𝜑 → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
19	2, 11, 12, 13, 10, 14, 15	yonffth 18397	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
20	18, 19	eqeltrrd 2861	. . . 4 ⊢ (𝜑 → ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21		eqid 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		yoniso.e	. . . . . 6 ⊢ 𝐸 = (𝑄 ↾_s ran (1^st ‘𝑌))
23	11	oppccat 17835	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
24	10, 23	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Cat)
25	12	setccat 18199	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑊 → 𝑆 ∈ Cat)
26	14, 25	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Cat)
27	13, 24, 26	fuccat 18087	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Cat)
28		fvex 6894	. . . . . . . 8 ⊢ (1^st ‘𝑌) ∈ V
29	28	rnex 7913	. . . . . . 7 ⊢ ran (1^st ‘𝑌) ∈ V
30	29	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (1^st ‘𝑌) ∈ V)
31	13	fucbas 18077	. . . . . . . . 9 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32		1st2ndbr 8044	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
33	1, 16, 32	sylancr 599	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
34	21, 31, 33	funcf1 17980	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
35	34	ffnd 6706	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌) Fn (Base‘𝐶))
36		dffn3 6718	. . . . . . 7 ⊢ ((1^st ‘𝑌) Fn (Base‘𝐶) ↔ (1^st ‘𝑌):(Base‘𝐶)⟶ran (1^st ‘𝑌))
37	35, 36	sylib 221	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)⟶ran (1^st ‘𝑌))
38	21, 22, 27, 30, 37	ffthres2c 18056	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2^nd ‘𝑌) ↔ (1^st ‘𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2^nd ‘𝑌)))
39		df-br 5104	. . . . 5 ⊢ ((1^st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2^nd ‘𝑌) ↔ ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
40		df-br 5104	. . . . 5 ⊢ ((1^st ‘𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2^nd ‘𝑌) ↔ ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
41	38, 39, 40	3bitr3g 316	. . . 4 ⊢ (𝜑 → (⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)) ↔ ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))))
42	20, 41	mpbid 235	. . 3 ⊢ (𝜑 → ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
43	18, 42	eqeltrd 2860	. 2 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
44		fveq2 6881	. . . . . . . . 9 ⊢ (((1^st ‘𝑌)‘𝑥) = ((1^st ‘𝑌)‘𝑦) → (1^st ‘((1^st ‘𝑌)‘𝑥)) = (1^st ‘((1^st ‘𝑌)‘𝑦)))
45	44	fveq1d 6883	. . . . . . . 8 ⊢ (((1^st ‘𝑌)‘𝑥) = ((1^st ‘𝑌)‘𝑦) → ((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥) = ((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥))
46	45	fveq2d 6885	. . . . . . 7 ⊢ (((1^st ‘𝑌)‘𝑥) = ((1^st ‘𝑌)‘𝑦) → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1^st ‘((1^st ‘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 2843	. . . . . . . . . . . . 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 6886	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (1^st ‘((1^st ‘𝑌)‘𝑦)) = (1^st ‘((1^st ‘𝑌)‘𝑥)))
53	52	fveq1d 6883	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥) = ((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥))
54	53	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥)))
55		id 23	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
56	54, 55	eqeq12d 2776	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦 ↔ (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥))
57	51, 56	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦) ↔ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1^st ‘((1^st ‘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 2760	. . . . . . . . . . . . 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 18377	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
63	62	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1^st ‘((1^st ‘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 2795	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦)
66	57, 65	chvarvv 2022	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
67	48, 66	sylan2 605	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
68	67, 65	eqeq12d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥)) ↔ 𝑥 = 𝑦))
69	46, 68	imbitrid 247	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1^st ‘𝑌)‘𝑥) = ((1^st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
70	69	ralrimivva 3205	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1^st ‘𝑌)‘𝑥) = ((1^st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
71		dff13 7254	. . . . 5 ⊢ ((1^st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆) ↔ ((1^st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1^st ‘𝑌)‘𝑥) = ((1^st ‘𝑌)‘𝑦) → 𝑥 = 𝑦)))
72	34, 70, 71	sylanbrc 595	. . . 4 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
73		f1f1orn 6832	. . . 4 ⊢ ((1^st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆) → (1^st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1^st ‘𝑌))
74	72, 73	syl 18	. . 3 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1^st ‘𝑌))
75	34	frnd 6714	. . . . 5 ⊢ (𝜑 → ran (1^st ‘𝑌) ⊆ (𝑂 Func 𝑆))
76	22, 31	ressbas2 17355	. . . . 5 ⊢ (ran (1^st ‘𝑌) ⊆ (𝑂 Func 𝑆) → ran (1^st ‘𝑌) = (Base‘𝐸))
77	75, 76	syl 18	. . . 4 ⊢ (𝜑 → ran (1^st ‘𝑌) = (Base‘𝐸))
78	77	f1oeq3d 6817	. . 3 ⊢ (𝜑 → ((1^st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1^st ‘𝑌) ↔ (1^st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸)))
79	74, 78	mpbid 235	. 2 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))
80		eqid 2760	. . 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 18225	. 2 ⊢ (𝜑 → (𝑌 ∈ (𝐶𝐼𝐸) ↔ (𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)) ∧ (1^st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))))
84	43, 79, 83	mpbir2and 726	1 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝐸))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 ⟨cop 4590 class class class wbr 5103 ran crn 5656 Rel wrel 5660 Fn wfn 6530 ⟶wf 6531 –1-1→wf1 6532 –1-1-onto→wf1o 6534 ‘cfv 6535 (class class class)co 7416 1^st c1st 7990 2^nd c2nd 7991 Basecbs 17326 ↾_s cress 17347 Hom chom 17378 Catccat 17777 Hom_f chomf 17779 oppCatcoppc 17824 Isociso 17860 Func cfunc 17968 Full cful 18018 Faith cfth 18019 FuncCat cfuc 18059 SetCatcsetc 18189 CatCatccatc 18212 Yoncyon 18362

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8229 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-fz 13587 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-hom 17391 df-cco 17392 df-cat 17781 df-cid 17782 df-homf 17783 df-comf 17784 df-oppc 17825 df-sect 17861 df-inv 17862 df-iso 17863 df-ssc 17924 df-resc 17925 df-subc 17926 df-func 17972 df-idfu 17973 df-cofu 17974 df-full 18020 df-fth 18021 df-nat 18060 df-fuc 18061 df-setc 18190 df-catc 18213 df-xpc 18285 df-1stf 18286 df-2ndf 18287 df-prf 18288 df-evlf 18326 df-curf 18327 df-hof 18363 df-yon 18364

This theorem is used by: (None)

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