Theorem yoniso 17133

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 16729	. . . 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 16954	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑉 ∩ Cat))
7		inss2 3982	. . . . . . 7 ⊢ (𝑉 ∩ Cat) ⊆ Cat
8	6, 7	syl6eqss 3804	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ Cat)
9		yoniso.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
10	8, 9	sseldd 3753	. . . . 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 17110	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
17		1st2nd 7363	. . . 4 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
18	1, 16, 17	sylancr 567	. . 3 ⊢ (𝜑 → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
19	2, 11, 12, 13, 10, 14, 15	yonffth 17132	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
20	18, 19	eqeltrrd 2851	. . . 4 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21		eqid 2771	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		yoniso.e	. . . . . 6 ⊢ 𝐸 = (𝑄 ↾s ran (1st ‘𝑌))
23	11	oppccat 16589	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
24	10, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Cat)
25	12	setccat 16942	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑊 → 𝑆 ∈ Cat)
26	14, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Cat)
27	13, 24, 26	fuccat 16837	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Cat)
28		fvex 6342	. . . . . . . 8 ⊢ (1st ‘𝑌) ∈ V
29	28	rnex 7247	. . . . . . 7 ⊢ ran (1st ‘𝑌) ∈ V
30	29	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (1st ‘𝑌) ∈ V)
31	13	fucbas 16827	. . . . . . . . 9 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32		1st2ndbr 7366	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
33	1, 16, 32	sylancr 567	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
34	21, 31, 33	funcf1 16733	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
35		ffn 6185	. . . . . . . 8 ⊢ ((1st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆) → (1st ‘𝑌) Fn (Base‘𝐶))
36	34, 35	syl 17	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌) Fn (Base‘𝐶))
37		dffn3 6194	. . . . . . 7 ⊢ ((1st ‘𝑌) Fn (Base‘𝐶) ↔ (1st ‘𝑌):(Base‘𝐶)⟶ran (1st ‘𝑌))
38	36, 37	sylib 208	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)⟶ran (1st ‘𝑌))
39	21, 22, 27, 30, 38	ffthres2c 16807	. . . . 5 ⊢ (𝜑 → ((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ (1st ‘𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd ‘𝑌)))
40		df-br 4787	. . . . 5 ⊢ ((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
41		df-br 4787	. . . . 5 ⊢ ((1st ‘𝑌)((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))(2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
42	39, 40, 41	3bitr3g 302	. . . 4 ⊢ (𝜑 → (⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))))
43	20, 42	mpbid 222	. . 3 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
44	18, 43	eqeltrd 2850	. 2 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
45		fveq2 6332	. . . . . . . . 9 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → (1st ‘((1st ‘𝑌)‘𝑥)) = (1st ‘((1st ‘𝑌)‘𝑦)))
46	45	fveq1d 6334	. . . . . . . 8 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → ((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥) = ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥))
47	46	fveq2d 6336	. . . . . . 7 ⊢ (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)))
48		simpl 468	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
49	48, 48	jca 495	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
50		eleq1w 2833	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
51	50	anbi2d 606	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
52	51	anbi2d 606	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
53		fveq2 6332	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((1st ‘𝑌)‘𝑦) = ((1st ‘𝑌)‘𝑥))
54	53	fveq2d 6336	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (1st ‘((1st ‘𝑌)‘𝑦)) = (1st ‘((1st ‘𝑌)‘𝑥)))
55	54	fveq1d 6334	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥) = ((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥))
56	55	fveq2d 6336	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)))
57		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
58	56, 57	eqeq12d 2786	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦 ↔ (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥))
59	52, 58	imbi12d 333	. . . . . . . . . 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)))
60	10	adantr 466	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
61		simprr 748	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
62		eqid 2771	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
63		simprl 746	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
64	2, 21, 60, 61, 62, 63	yon11 17112	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
65	64	fveq2d 6336	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = (𝐹‘(𝑥(Hom ‘𝐶)𝑦)))
66		yoniso.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘(𝑥(Hom ‘𝐶)𝑦)) = 𝑦)
67	65, 66	eqtrd 2805	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦)
68	59, 67	chvarv 2425	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
69	49, 68	sylan2 572	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
70	69, 67	eqeq12d 2786	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝐹‘((1st ‘((1st ‘𝑌)‘𝑥))‘𝑥)) = (𝐹‘((1st ‘((1st ‘𝑌)‘𝑦))‘𝑥)) ↔ 𝑥 = 𝑦))
71	47, 70	syl5ib 234	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
72	71	ralrimivva 3120	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1st ‘𝑌)‘𝑥) = ((1st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
73		dff13 6655	. . . . 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)))
74	34, 72, 73	sylanbrc 564	. . . 4 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
75		f1f1orn 6289	. . . 4 ⊢ ((1st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆) → (1st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1st ‘𝑌))
76	74, 75	syl 17	. . 3 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1st ‘𝑌))
77		frn 6193	. . . . . 6 ⊢ ((1st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆) → ran (1st ‘𝑌) ⊆ (𝑂 Func 𝑆))
78	34, 77	syl 17	. . . . 5 ⊢ (𝜑 → ran (1st ‘𝑌) ⊆ (𝑂 Func 𝑆))
79	22, 31	ressbas2 16138	. . . . 5 ⊢ (ran (1st ‘𝑌) ⊆ (𝑂 Func 𝑆) → ran (1st ‘𝑌) = (Base‘𝐸))
80	78, 79	syl 17	. . . 4 ⊢ (𝜑 → ran (1st ‘𝑌) = (Base‘𝐸))
81		f1oeq3 6270	. . . 4 ⊢ (ran (1st ‘𝑌) = (Base‘𝐸) → ((1st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1st ‘𝑌) ↔ (1st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸)))
82	80, 81	syl 17	. . 3 ⊢ (𝜑 → ((1st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1st ‘𝑌) ↔ (1st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸)))
83	76, 82	mpbid 222	. 2 ⊢ (𝜑 → (1st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))
84		eqid 2771	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
85		yoniso.eb	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
86		yoniso.i	. . 3 ⊢ 𝐼 = (Iso‘𝐷)
87	3, 4, 21, 84, 5, 9, 85, 86	catciso 16964	. 2 ⊢ (𝜑 → (𝑌 ∈ (𝐶𝐼𝐸) ↔ (𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)) ∧ (1st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))))
88	44, 83, 87	mpbir2and 684	1 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝐸))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723  ⟨cop 4322   class class class wbr 4786  ran crn 5250  Rel wrel 5254   Fn wfn 6026  ⟶wf 6027  –1-1→wf1 6028  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6793  1^st c1st 7313  2^nd c2nd 7314  Basecbs 16064   ↾_s cress 16065  Hom chom 16160  Catccat 16532  Hom_f chomf 16534  oppCatcoppc 16578  Isociso 16613   Func cfunc 16721   Full cful 16769   Faith cfth 16770   FuncCat cfuc 16809  SetCatcsetc 16932  CatCatccatc 16951  Yoncyon 17097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-tpos 7504  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-hom 16174  df-cco 16175  df-cat 16536  df-cid 16537  df-homf 16538  df-comf 16539  df-oppc 16579  df-sect 16614  df-inv 16615  df-iso 16616  df-ssc 16677  df-resc 16678  df-subc 16679  df-func 16725  df-idfu 16726  df-cofu 16727  df-full 16771  df-fth 16772  df-nat 16810  df-fuc 16811  df-setc 16933  df-catc 16952  df-xpc 17020  df-1stf 17021  df-2ndf 17022  df-prf 17023  df-evlf 17061  df-curf 17062  df-hof 17098  df-yon 17099

This theorem is referenced by: (None)