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Theorem opnregcld 33791
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 ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
Distinct variable groups:   𝐴,𝑜   𝑜,𝐽   𝑜,𝑋

Proof of Theorem opnregcld
StepHypRef Expression
1 opnregcld.1 . . . . 5 𝑋 = 𝐽
21ntropn 21654 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
3 eqcom 2805 . . . . 5 (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
43biimpi 219 . . . 4 (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
5 fveq2 6645 . . . . 5 (𝑜 = ((int‘𝐽)‘𝐴) → ((cls‘𝐽)‘𝑜) = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
65rspceeqv 3586 . . . 4 ((((int‘𝐽)‘𝐴) ∈ 𝐽𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴))) → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
72, 4, 6syl2an 598 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴) → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
87ex 416 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
9 simpl 486 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝐽 ∈ Top)
101eltopss 21512 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝑋)
111clsss3 21664 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝑋) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
1210, 11syldan 594 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
131ntrss2 21662 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
1412, 13syldan 594 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
151clsss 21659 . . . . . . . 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)))
169, 12, 14, 15syl3anc 1368 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)))
171clsidm 21672 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
1810, 17syldan 594 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
1916, 18sseqtrd 3955 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘𝑜))
201ntrss3 21665 . . . . . . . 8 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
2112, 20syldan 594 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
22 simpr 488 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝐽)
231sscls 21661 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝑋) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
2410, 23syldan 594 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
251ssntr 21663 . . . . . . . 8 (((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) ∧ (𝑜𝐽𝑜 ⊆ ((cls‘𝐽)‘𝑜))) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
269, 12, 22, 24, 25syl22anc 837 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
271clsss 21659 . . . . . . 7 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜))) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
289, 21, 26, 27syl3anc 1368 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
2919, 28eqssd 3932 . . . . 5 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
3029adantlr 714 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
31 2fveq3 6650 . . . . 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))
3331, 32eqeq12d 2814 . . . 4 (𝐴 = ((cls‘𝐽)‘𝑜) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜)))
3430, 33syl5ibrcom 250 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
3534rexlimdva 3243 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
368, 35impbid 215 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wrex 3107  wss 3881   cuni 4800  cfv 6324  Topctop 21498  intcnt 21622  clsccl 21623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-top 21499  df-cld 21624  df-ntr 21625  df-cls 21626
This theorem is referenced by: (None)
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