Theorem opnregcld 31495

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 20663	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
3		eqcom 2617	. . . . 5 ⊢ (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
4	3	biimpi 205	. . . 4 ⊢ (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
5		fveq2 6103	. . . . . 6 ⊢ (𝑜 = ((int‘𝐽)‘𝐴) → ((cls‘𝐽)‘𝑜) = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
6	5	eqeq2d 2620	. . . . 5 ⊢ (𝑜 = ((int‘𝐽)‘𝐴) → (𝐴 = ((cls‘𝐽)‘𝑜) ↔ 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴))))
7	6	rspcev 3282	. . . 4 ⊢ ((((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴))) → ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
8	2, 4, 7	syl2an 493	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴) → ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
9	8	ex 449	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
10		simpl 472	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝐽 ∈ Top)
11	1	eltopss 20537	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
12	1	clsss3 20673	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
13	11, 12	syldan 486	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
14	1	ntrss2 20671	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
15	13, 14	syldan 486	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
16	1	clsss 20668	. . . . . . . 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)))
17	10, 13, 15, 16	syl3anc 1318	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)))
18	1	clsidm 20681	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
19	11, 18	syldan 486	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
20	17, 19	sseqtrd 3604	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘𝑜))
21	1	ntrss3 20674	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
22	13, 21	syldan 486	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
23		simpr 476	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ∈ 𝐽)
24	1	sscls 20670	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
25	11, 24	syldan 486	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
26	1	ssntr 20672	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ ((cls‘𝐽)‘𝑜))) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
27	10, 13, 23, 25, 26	syl22anc 1319	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
28	1	clsss 20668	. . . . . . 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))))
29	10, 22, 27, 28	syl3anc 1318	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
30	20, 29	eqssd 3585	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
31	30	adantlr 747	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
32		fveq2 6103	. . . . . 6 ⊢ (𝐴 = ((cls‘𝐽)‘𝑜) → ((int‘𝐽)‘𝐴) = ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
33	32	fveq2d 6107	. . . . 5 ⊢ (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
34		id 22	. . . . 5 ⊢ (𝐴 = ((cls‘𝐽)‘𝑜) → 𝐴 = ((cls‘𝐽)‘𝑜))
35	33, 34	eqeq12d 2625	. . . 4 ⊢ (𝐴 = ((cls‘𝐽)‘𝑜) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜)))
36	31, 35	syl5ibrcom 236	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
37	36	rexlimdva 3013	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
38	9, 37	impbid 201	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  intcnt 20631  clsccl 20632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-ntr 20634  df-cls 20635

This theorem is referenced by: (None)