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Theorem yoniso 18242

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 17816	. . . 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 18055	. . . . . . 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 18219	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
17		1st2nd 8027	. . . 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 18241	. . . . 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 17672	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
24	10, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Cat)
25	12	setccat 18039	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑊 → 𝑆 ∈ Cat)
26	14, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Cat)
27	13, 24, 26	fuccat 17927	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Cat)
28		fvex 6904	. . . . . . . 8 ⊢ (1^st ‘𝑌) ∈ V
29	28	rnex 7905	. . . . . . 7 ⊢ ran (1^st ‘𝑌) ∈ V
30	29	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (1^st ‘𝑌) ∈ V)
31	13	fucbas 17916	. . . . . . . . 9 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32		1st2ndbr 8030	. . . . . . . . . 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 17820	. . . . . . . 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 17895	. . . . 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 18221	. . . . . . . . . . . 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 7256	. . . . 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 17186	. . . . 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 18065	. 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 7411 1^st c1st 7975 2^nd c2nd 7976 Basecbs 17148 ↾_s cress 17177 Hom chom 17212 Catccat 17612 Hom_f chomf 17614 oppCatcoppc 17659 Isociso 17697 Func cfunc 17808 Full cful 17857 Faith cfth 17858 FuncCat cfuc 17897 SetCatcsetc 18029 CatCatccatc 18052 Yoncyon 18206

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 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-hom 17225 df-cco 17226 df-cat 17616 df-cid 17617 df-homf 17618 df-comf 17619 df-oppc 17660 df-sect 17698 df-inv 17699 df-iso 17700 df-ssc 17761 df-resc 17762 df-subc 17763 df-func 17812 df-idfu 17813 df-cofu 17814 df-full 17859 df-fth 17860 df-nat 17898 df-fuc 17899 df-setc 18030 df-catc 18053 df-xpc 18128 df-1stf 18129 df-2ndf 18130 df-prf 18131 df-evlf 18170 df-curf 18171 df-hof 18207 df-yon 18208

This theorem is referenced by: (None)

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