Theorem cldregopn 36842

Description: A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)

Hypothesis

Ref	Expression
opnregcld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cldregopn	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))

Distinct variable groups:   𝐴,𝑐   𝐽,𝑐   𝑋,𝑐

Proof of Theorem cldregopn

Step	Hyp	Ref	Expression
1		opnregcld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	clscld 23204	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽))
3		eqcom 2770	. . . . 5 ⊢ (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
4	3	biimpi 219	. . . 4 ⊢ (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
5		fveq2 6881	. . . . 5 ⊢ (𝑐 = ((cls‘𝐽)‘𝐴) → ((int‘𝐽)‘𝑐) = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
6	5	rspceeqv 3604	. . . 4 ⊢ ((((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
7	2, 4, 6	syl2an 607	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
8	7	ex 417	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
9		cldrcl 23183	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
10	1	cldss 23186	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝑐 ⊆ 𝑋)
11	1	ntrss2 23214	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
12	9, 10, 11	syl2anc 595	. . . . . . . 8 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
13	1	clsss2 23229	. . . . . . . 8 ⊢ ((𝑐 ∈ (Clsd‘𝐽) ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑐) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
14	12, 13	mpdan 699	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
15	1	ntrss 23212	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
16	9, 10, 14, 15	syl3anc 1398	. . . . . 6 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
17	1	ntridm 23225	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
18	9, 10, 17	syl2anc 595	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
19	1	ntrss3 23217	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
20	9, 10, 19	syl2anc 595	. . . . . . . . 9 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
21	1	clsss3 23216	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
22	9, 20, 21	syl2anc 595	. . . . . . . 8 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
23	1	sscls 23213	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
24	9, 20, 23	syl2anc 595	. . . . . . . 8 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
25	1	ntrss 23212	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐))) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
26	9, 22, 24, 25	syl3anc 1398	. . . . . . 7 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
27	18, 26	eqsstrrd 3972	. . . . . 6 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
28	16, 27	eqssd 3954	. . . . 5 ⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
29	28	adantl 486	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
30		2fveq3 6886	. . . . 5 ⊢ (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
31		id 23	. . . . 5 ⊢ (𝐴 = ((int‘𝐽)‘𝑐) → 𝐴 = ((int‘𝐽)‘𝑐))
32	30, 31	eqeq12d 2779	. . . 4 ⊢ (𝐴 = ((int‘𝐽)‘𝑐) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐)))
33	29, 32	syl5ibrcom 250	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
34	33	rexlimdva 3166	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
35	8, 34	impbid 215	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ⊆ wss 3905  ∪ cuni 4872  ‘cfv 6536  Topctop 23050  Clsdccld 23173  intcnt 23174  clsccl 23175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 23051  df-cld 23176  df-ntr 23177  df-cls 23178

This theorem is referenced by: (None)