Theorem opnregcld 36374

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

Hypothesis

Ref	Expression
opnregcld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
opnregcld	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))

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

Proof of Theorem opnregcld

Step	Hyp	Ref	Expression
1		opnregcld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	ntropn 22964	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
3		eqcom 2738	. . . . 5 ⊢ (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
4	3	biimpi 216	. . . 4 ⊢ (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
5		fveq2 6822	. . . . 5 ⊢ (𝑜 = ((int‘𝐽)‘𝐴) → ((cls‘𝐽)‘𝑜) = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
6	5	rspceeqv 3595	. . . 4 ⊢ ((((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴))) → ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
7	2, 4, 6	syl2an 596	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴) → ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
8	7	ex 412	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
9		simpl 482	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝐽 ∈ Top)
10	1	eltopss 22822	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
11	1	clsss3 22974	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
12	10, 11	syldan 591	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
13	1	ntrss2 22972	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
14	12, 13	syldan 591	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
15	1	clsss 22969	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜)) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)))
16	9, 12, 14, 15	syl3anc 1373	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)))
17	1	clsidm 22982	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
18	10, 17	syldan 591	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
19	16, 18	sseqtrd 3966	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘𝑜))
20	1	ntrss3 22975	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
21	12, 20	syldan 591	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
22		simpr 484	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ∈ 𝐽)
23	1	sscls 22971	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
24	10, 23	syldan 591	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
25	1	ssntr 22973	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ ((cls‘𝐽)‘𝑜))) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
26	9, 12, 22, 24, 25	syl22anc 838	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
27	1	clsss 22969	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋 ∧ 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜))) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
28	9, 21, 26, 27	syl3anc 1373	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
29	19, 28	eqssd 3947	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
30	29	adantlr 715	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
31		2fveq3 6827	. . . . 5 ⊢ (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
32		id 22	. . . . 5 ⊢ (𝐴 = ((cls‘𝐽)‘𝑜) → 𝐴 = ((cls‘𝐽)‘𝑜))
33	31, 32	eqeq12d 2747	. . . 4 ⊢ (𝐴 = ((cls‘𝐽)‘𝑜) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜)))
34	30, 33	syl5ibrcom 247	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
35	34	rexlimdva 3133	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
36	8, 35	impbid 212	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3897  ∪ cuni 4856  ‘cfv 6481  Topctop 22808  intcnt 22932  clsccl 22933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-top 22809  df-cld 22934  df-ntr 22935  df-cls 22936

This theorem is referenced by: (None)