Theorem opnregcld 36869

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 23217	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
3		eqcom 2769	. . . . 5 ⊢ (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
4	3	biimpi 219	. . . 4 ⊢ (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
5		fveq2 6881	. . . . 5 ⊢ (𝑜 = ((int‘𝐽)‘𝐴) → ((cls‘𝐽)‘𝑜) = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
6	5	rspceeqv 3603	. . . 4 ⊢ ((((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ 𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴))) → ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
7	2, 4, 6	syl2an 607	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴) → ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
8	7	ex 417	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
9		simpl 487	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝐽 ∈ Top)
10	1	eltopss 23075	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
11	1	clsss3 23227	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
12	10, 11	syldan 602	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
13	1	ntrss2 23225	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
14	12, 13	syldan 602	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
15	1	clsss 23222	. . . . . . . 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 1397	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)))
17	1	clsidm 23235	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
18	10, 17	syldan 602	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
19	16, 18	sseqtrd 3972	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘𝑜))
20	1	ntrss3 23228	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
21	12, 20	syldan 602	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
22		simpr 489	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ∈ 𝐽)
23	1	sscls 23224	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ 𝑋) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
24	10, 23	syldan 602	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
25	1	ssntr 23226	. . . . . . . 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 851	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
27	1	clsss 23222	. . . . . . 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 1397	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
29	19, 28	eqssd 3953	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
30	29	adantlr 727	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
31		2fveq3 6886	. . . . 5 ⊢ (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
32		id 23	. . . . 5 ⊢ (𝐴 = ((cls‘𝐽)‘𝑜) → 𝐴 = ((cls‘𝐽)‘𝑜))
33	31, 32	eqeq12d 2778	. . . 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 250	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
35	34	rexlimdva 3165	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
36	8, 35	impbid 215	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜 ∈ 𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   ⊆ wss 3904  ∪ cuni 4871  ‘cfv 6536  Topctop 23061  intcnt 23185  clsccl 23186

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  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 23062  df-cld 23187  df-ntr 23188  df-cls 23189

This theorem is used by: (None)