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Theorem dvcnvrelem2 23781

Description: Lemma for dvcnvre 23782. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

**Hypotheses**
Ref	Expression
dvcnvre.f	⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
dvcnvre.d	⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
dvcnvre.z	⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
dvcnvre.1	⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
dvcnvre.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
dvcnvre.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
dvcnvre.s	⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
dvcnvre.t	⊢ 𝑇 = (topGen‘ran (,))
dvcnvre.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
dvcnvre.m	⊢ 𝑀 = (𝐽 ↾_t 𝑋)
dvcnvre.n	⊢ 𝑁 = (𝐽 ↾_t 𝑌)

**Assertion**
Ref	Expression
dvcnvrelem2	⊢ (𝜑 → ((𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌) ∧ ^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶))))

**Proof of Theorem dvcnvrelem2**
Step	Hyp	Ref	Expression
1		dvcnvre.t	. . . . . 6 ⊢ 𝑇 = (topGen‘ran (,))
2		retop 22565	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
3	1, 2	eqeltri 2697	. . . . 5 ⊢ 𝑇 ∈ Top
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Top)
5		dvcnvre.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
6		f1ofo 6144	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
7		forn 6118	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
8	5, 6, 7	3syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐹 = 𝑌)
9		dvcnvre.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
10		cncff 22696	. . . . . 6 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
11		frn 6053	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
12	9, 10, 11	3syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
13	8, 12	eqsstr3d 3640	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
14		imassrn 5477	. . . . 5 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ran 𝐹
15	14, 8	syl5sseq 3653	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌)
16		uniretop 22566	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
17	1	unieqi 4445	. . . . . 6 ⊢ ∪ 𝑇 = ∪ (topGen‘ran (,))
18	16, 17	eqtr4i 2647	. . . . 5 ⊢ ℝ = ∪ 𝑇
19	18	ntrss 20859	. . . 4 ⊢ ((𝑇 ∈ Top ∧ 𝑌 ⊆ ℝ ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌) → ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ((int‘𝑇)‘𝑌))
20	4, 13, 15, 19	syl3anc 1326	. . 3 ⊢ (𝜑 → ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ((int‘𝑇)‘𝑌))
21		dvcnvre.d	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
22		dvcnvre.z	. . . . 5 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
23		dvcnvre.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
24		dvcnvre.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
25		dvcnvre.s	. . . . 5 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
26	9, 21, 22, 5, 23, 24, 25	dvcnvrelem1 23780	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
27	1	fveq2i 6194	. . . . 5 ⊢ (int‘𝑇) = (int‘(topGen‘ran (,)))
28	27	fveq1i 6192	. . . 4 ⊢ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
29	26, 28	syl6eleqr 2712	. . 3 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
30	20, 29	sseldd 3604	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌))
31		f1ocnv 6149	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
32		f1of 6137	. . . . . . 7 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
33	5, 31, 32	3syl 18	. . . . . 6 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
34		ffun 6048	. . . . . 6 ⊢ (^◡𝐹:𝑌⟶𝑋 → Fun ^◡𝐹)
35		funcnvres 5967	. . . . . 6 ⊢ (Fun ^◡𝐹 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
36	33, 34, 35	3syl 18	. . . . 5 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
37		dvbsss 23666	. . . . . . . . . . 11 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
38	21, 37	syl6eqssr 3656	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
39		ax-resscn 9993	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
40	38, 39	syl6ss 3615	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
41		cncfss 22702	. . . . . . . . 9 ⊢ ((((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋 ∧ 𝑋 ⊆ ℂ) → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
42	25, 40, 41	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
43		f1of1 6136	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
44	5, 43	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌)
45		f1ores 6151	. . . . . . . . . 10 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋) → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
46	44, 25, 45	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
47		dvcnvre.j	. . . . . . . . . . . . . . 15 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
48	47	tgioo2 22606	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
49	1, 48	eqtri 2644	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝐽 ↾_t ℝ)
50	49	oveq1i 6660	. . . . . . . . . . . 12 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
51	47	cnfldtop 22587	. . . . . . . . . . . . . 14 ⊢ 𝐽 ∈ Top
52	51	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
53	25, 38	sstrd 3613	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
54		reex 10027	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
55	54	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
56		restabs 20969	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ ∧ ℝ ∈ V) → ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
57	52, 53, 55, 56	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
58	50, 57	syl5eq 2668	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
59	38, 23	sseldd 3604	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
60	24	rpred 11872	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
61	59, 60	resubcld 10458	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
62	59, 60	readdcld 10069	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
63		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
64	1, 63	icccmp 22628	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
65	61, 62, 64	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
66	58, 65	eqeltrrd 2702	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
67		f1of 6137	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))⟶(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
68	46, 67	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))⟶(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
69	12, 39	syl6ss 3615	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
70	14, 69	syl5ss 3614	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ)
71		rescncf 22700	. . . . . . . . . . . . 13 ⊢ (((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋 → (𝐹 ∈ (𝑋–cn→ℝ) → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ)))
72	25, 9, 71	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ))
73		cncffvrn 22701	. . . . . . . . . . . 12 ⊢ (((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ ∧ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ)) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ↔ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))⟶(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
74	70, 72, 73	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ↔ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))⟶(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
75	68, 74	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
76		eqid 2622	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
77	47, 76	cncfcnvcn 22724	. . . . . . . . . 10 ⊢ (((𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp ∧ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ↔ ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
78	66, 75, 77	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ↔ ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
79	46, 78	mpbid 222	. . . . . . . 8 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
80	42, 79	sseldd 3604	. . . . . . 7 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
81		eqid 2622	. . . . . . . . 9 ⊢ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
82		dvcnvre.m	. . . . . . . . 9 ⊢ 𝑀 = (𝐽 ↾_t 𝑋)
83	47, 81, 82	cncfcn 22712	. . . . . . . 8 ⊢ (((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ ∧ 𝑋 ⊆ ℂ) → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
84	70, 40, 83	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
85	80, 84	eleqtrd 2703	. . . . . 6 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
86	59, 24	ltsubrpd 11904	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
87	61, 59, 86	ltled 10185	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
88	59, 24	ltaddrpd 11905	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
89	59, 62, 88	ltled 10185	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
90		elicc2 12238	. . . . . . . . . 10 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
91	61, 62, 90	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
92	59, 87, 89, 91	mpbir3and 1245	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
93		ffun 6048	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶ℝ → Fun 𝐹)
94	9, 10, 93	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
95		fdm 6051	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶ℝ → dom 𝐹 = 𝑋)
96	9, 10, 95	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑋)
97	25, 96	sseqtr4d 3642	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
98		funfvima2 6493	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘𝐶) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
99	94, 97, 98	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘𝐶) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
100	92, 99	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
101	47	cnfldtopon 22586	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
102		resttopon 20965	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ) → (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (TopOn‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
103	101, 70, 102	sylancr 695	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (TopOn‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
104		toponuni 20719	. . . . . . . 8 ⊢ ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (TopOn‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
105	103, 104	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
106	100, 105	eleqtrd 2703	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
107		eqid 2622	. . . . . . 7 ⊢ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
108	107	cncnpi 21082	. . . . . 6 ⊢ ((^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀) ∧ (𝐹‘𝐶) ∈ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))) → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
109	85, 106, 108	syl2anc 693	. . . . 5 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
110	36, 109	eqeltrrd 2702	. . . 4 ⊢ (𝜑 → (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
111		dvcnvre.n	. . . . . . . 8 ⊢ 𝑁 = (𝐽 ↾_t 𝑌)
112	111	oveq1i 6660	. . . . . . 7 ⊢ (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
113		ssexg 4804	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℝ ∧ ℝ ∈ V) → 𝑌 ∈ V)
114	13, 54, 113	sylancl 694	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ V)
115		restabs 20969	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌 ∧ 𝑌 ∈ V) → ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
116	52, 15, 114, 115	syl3anc 1326	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
117	112, 116	syl5eq 2668	. . . . . 6 ⊢ (𝜑 → (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
118	117	oveq1d 6665	. . . . 5 ⊢ (𝜑 → ((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀))
119	118	fveq1d 6193	. . . 4 ⊢ (𝜑 → (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)) = (((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
120	110, 119	eleqtrrd 2704	. . 3 ⊢ (𝜑 → (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
121	13, 39	syl6ss 3615	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
122		resttopon 20965	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑌 ⊆ ℂ) → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
123	101, 121, 122	sylancr 695	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
124	111, 123	syl5eqel 2705	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘𝑌))
125		topontop 20718	. . . . 5 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑁 ∈ Top)
126	124, 125	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Top)
127		toponuni 20719	. . . . . 6 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑁)
128	124, 127	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝑁)
129	15, 128	sseqtrd 3641	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ∪ 𝑁)
130	15, 13	sstrd 3613	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℝ)
131		difssd 3738	. . . . . . . . 9 ⊢ (𝜑 → (ℝ ∖ 𝑌) ⊆ ℝ)
132	130, 131	unssd 3789	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌)) ⊆ ℝ)
133		ssun1 3776	. . . . . . . . 9 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))
134	133	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌)))
135	18	ntrss 20859	. . . . . . . 8 ⊢ ((𝑇 ∈ Top ∧ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌)) ⊆ ℝ ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) → ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
136	4, 132, 134, 135	syl3anc 1326	. . . . . . 7 ⊢ (𝜑 → ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
137	136, 29	sseldd 3604	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
138		f1of 6137	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
139	5, 138	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
140	139, 23	ffvelrnd 6360	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌)
141	137, 140	elind 3798	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
142		eqid 2622	. . . . . . . 8 ⊢ (𝑇 ↾_t 𝑌) = (𝑇 ↾_t 𝑌)
143	18, 142	restntr 20986	. . . . . . 7 ⊢ ((𝑇 ∈ Top ∧ 𝑌 ⊆ ℝ ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌) → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
144	4, 13, 15, 143	syl3anc 1326	. . . . . 6 ⊢ (𝜑 → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
145		restabs 20969	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ℝ ∧ ℝ ∈ V) → ((𝐽 ↾_t ℝ) ↾_t 𝑌) = (𝐽 ↾_t 𝑌))
146	52, 13, 55, 145	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → ((𝐽 ↾_t ℝ) ↾_t 𝑌) = (𝐽 ↾_t 𝑌))
147	49	oveq1i 6660	. . . . . . . . 9 ⊢ (𝑇 ↾_t 𝑌) = ((𝐽 ↾_t ℝ) ↾_t 𝑌)
148	146, 147, 111	3eqtr4g 2681	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ↾_t 𝑌) = 𝑁)
149	148	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (int‘(𝑇 ↾_t 𝑌)) = (int‘𝑁))
150	149	fveq1d 6193	. . . . . 6 ⊢ (𝜑 → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
151	144, 150	eqtr3d 2658	. . . . 5 ⊢ (𝜑 → (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
152	141, 151	eleqtrd 2703	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
153	128	feq2d 6031	. . . . . 6 ⊢ (𝜑 → (^◡𝐹:𝑌⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶𝑋))
154	33, 153	mpbid 222	. . . . 5 ⊢ (𝜑 → ^◡𝐹:∪ 𝑁⟶𝑋)
155		resttopon 20965	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑋 ⊆ ℂ) → (𝐽 ↾_t 𝑋) ∈ (TopOn‘𝑋))
156	101, 40, 155	sylancr 695	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾_t 𝑋) ∈ (TopOn‘𝑋))
157	82, 156	syl5eqel 2705	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑋))
158		toponuni 20719	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑀)
159		feq3 6028	. . . . . 6 ⊢ (𝑋 = ∪ 𝑀 → (^◡𝐹:∪ 𝑁⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶∪ 𝑀))
160	157, 158, 159	3syl 18	. . . . 5 ⊢ (𝜑 → (^◡𝐹:∪ 𝑁⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶∪ 𝑀))
161	154, 160	mpbid 222	. . . 4 ⊢ (𝜑 → ^◡𝐹:∪ 𝑁⟶∪ 𝑀)
162		eqid 2622	. . . . 5 ⊢ ∪ 𝑁 = ∪ 𝑁
163		eqid 2622	. . . . 5 ⊢ ∪ 𝑀 = ∪ 𝑀
164	162, 163	cnprest 21093	. . . 4 ⊢ (((𝑁 ∈ Top ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ∪ 𝑁) ∧ ((𝐹‘𝐶) ∈ ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ^◡𝐹:∪ 𝑁⟶∪ 𝑀)) → (^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶)) ↔ (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶))))
165	126, 129, 152, 161, 164	syl22anc 1327	. . 3 ⊢ (𝜑 → (^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶)) ↔ (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶))))
166	120, 165	mpbird 247	. 2 ⊢ (𝜑 → ^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶)))
167	30, 166	jca 554	1 ⊢ (𝜑 → ((𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌) ∧ ^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∪ cuni 4436 class class class wbr 4653 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 ≤ cle 10075 − cmin 10266 ℝ⁺crp 11832 (,)cioo 12175 [,]cicc 12178 ↾_t crest 16081 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 Topctop 20698 TopOnctopon 20715 intcnt 20821 Cn ccn 21028 CnP ccnp 21029 Compccmp 21189 –cn→ccncf 22679 D cdv 23627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: dvcnvre 23782

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