yoniso - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > yoniso		Structured version Visualization version GIF version

Theorem yoniso 18237

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 17811	. . . 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 18050	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑉 ∩ Cat))
7		inss2 4229	. . . . . . 7 ⊢ (𝑉 ∩ Cat) ⊆ Cat
8	6, 7	eqsstrdi 4036	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ Cat)
9		yoniso.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
10	8, 9	sseldd 3983	. . . . 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 18214	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
17		1st2nd 8024	. . . 4 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
18	1, 16, 17	sylancr 587	. . 3 ⊢ (𝜑 → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
19	2, 11, 12, 13, 10, 14, 15	yonffth 18236	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
20	18, 19	eqeltrrd 2834	. . . 4 ⊢ (𝜑 → ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
21		eqid 2732	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		yoniso.e	. . . . . 6 ⊢ 𝐸 = (𝑄 ↾_s ran (1^st ‘𝑌))
23	11	oppccat 17667	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
24	10, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Cat)
25	12	setccat 18034	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑊 → 𝑆 ∈ Cat)
26	14, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Cat)
27	13, 24, 26	fuccat 17922	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Cat)
28		fvex 6904	. . . . . . . 8 ⊢ (1^st ‘𝑌) ∈ V
29	28	rnex 7902	. . . . . . 7 ⊢ ran (1^st ‘𝑌) ∈ V
30	29	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (1^st ‘𝑌) ∈ V)
31	13	fucbas 17911	. . . . . . . . 9 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32		1st2ndbr 8027	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
33	1, 16, 32	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
34	21, 31, 33	funcf1 17815	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)⟶(𝑂 Func 𝑆))
35	34	ffnd 6718	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝑌) Fn (Base‘𝐶))
36		dffn3 6730	. . . . . . 7 ⊢ ((1^st ‘𝑌) Fn (Base‘𝐶) ↔ (1^st ‘𝑌):(Base‘𝐶)⟶ran (1^st ‘𝑌))
37	35, 36	sylib 217	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)⟶ran (1^st ‘𝑌))
38	21, 22, 27, 30, 37	ffthres2c 17890	. . . . 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 5149	. . . . 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 5149	. . . . 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 312	. . . 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 231	. . 3 ⊢ (𝜑 → ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
43	18, 42	eqeltrd 2833	. 2 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
44		fveq2 6891	. . . . . . . . 9 ⊢ (((1^st ‘𝑌)‘𝑥) = ((1^st ‘𝑌)‘𝑦) → (1^st ‘((1^st ‘𝑌)‘𝑥)) = (1^st ‘((1^st ‘𝑌)‘𝑦)))
45	44	fveq1d 6893	. . . . . . . 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 6895	. . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
48	47, 47	jca 512	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
49		eleq1w 2816	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐶) ↔ 𝑥 ∈ (Base‘𝐶)))
50	49	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))))
51	50	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ↔ (𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))))
52		2fveq3 6896	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (1^st ‘((1^st ‘𝑌)‘𝑦)) = (1^st ‘((1^st ‘𝑌)‘𝑥)))
53	52	fveq1d 6893	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥) = ((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥))
54	53	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥)) = (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥)))
55		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
56	54, 55	eqeq12d 2748	. . . . . . . . . . 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 344	. . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
59		simprr 771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
60		eqid 2732	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
61		simprl 769	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
62	2, 21, 58, 59, 60, 61	yon11 18216	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥) = (𝑥(Hom ‘𝐶)𝑦))
63	62	fveq2d 6895	. . . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑦))‘𝑥)) = 𝑦)
66	57, 65	chvarvv 2002	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
67	48, 66	sylan2 593	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘((1^st ‘((1^st ‘𝑌)‘𝑥))‘𝑥)) = 𝑥)
68	67, 65	eqeq12d 2748	. . . . . . 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 243	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1^st ‘𝑌)‘𝑥) = ((1^st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
70	69	ralrimivva 3200	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(((1^st ‘𝑌)‘𝑥) = ((1^st ‘𝑌)‘𝑦) → 𝑥 = 𝑦))
71		dff13 7253	. . . . 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 583	. . . 4 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)–1-1→(𝑂 Func 𝑆))
73		f1f1orn 6844	. . . 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 17	. . 3 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)–1-1-onto→ran (1^st ‘𝑌))
75	34	frnd 6725	. . . . 5 ⊢ (𝜑 → ran (1^st ‘𝑌) ⊆ (𝑂 Func 𝑆))
76	22, 31	ressbas2 17181	. . . . 5 ⊢ (ran (1^st ‘𝑌) ⊆ (𝑂 Func 𝑆) → ran (1^st ‘𝑌) = (Base‘𝐸))
77	75, 76	syl 17	. . . 4 ⊢ (𝜑 → ran (1^st ‘𝑌) = (Base‘𝐸))
78	77	f1oeq3d 6830	. . 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 231	. 2 ⊢ (𝜑 → (1^st ‘𝑌):(Base‘𝐶)–1-1-onto→(Base‘𝐸))
80		eqid 2732	. . 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 18060	. 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 711	1 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐼𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 ⟨cop 4634 class class class wbr 5148 ran crn 5677 Rel wrel 5681 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7408 1^st c1st 7972 2^nd c2nd 7973 Basecbs 17143 ↾_s cress 17172 Hom chom 17207 Catccat 17607 Hom_f chomf 17609 oppCatcoppc 17654 Isociso 17692 Func cfunc 17803 Full cful 17852 Faith cfth 17853 FuncCat cfuc 17892 SetCatcsetc 18024 CatCatccatc 18047 Yoncyon 18201

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-hom 17220 df-cco 17221 df-cat 17611 df-cid 17612 df-homf 17613 df-comf 17614 df-oppc 17655 df-sect 17693 df-inv 17694 df-iso 17695 df-ssc 17756 df-resc 17757 df-subc 17758 df-func 17807 df-idfu 17808 df-cofu 17809 df-full 17854 df-fth 17855 df-nat 17893 df-fuc 17894 df-setc 18025 df-catc 18048 df-xpc 18123 df-1stf 18124 df-2ndf 18125 df-prf 18126 df-evlf 18165 df-curf 18166 df-hof 18202 df-yon 18203

This theorem is referenced by: (None)

W3C validator