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Theorem opnregcld 33575
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 21585 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
3 eqcom 2825 . . . . 5 (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
43biimpi 217 . . . 4 (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
5 fveq2 6663 . . . . 5 (𝑜 = ((int‘𝐽)‘𝐴) → ((cls‘𝐽)‘𝑜) = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
65rspceeqv 3635 . . . 4 ((((int‘𝐽)‘𝐴) ∈ 𝐽𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴))) → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
72, 4, 6syl2an 595 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴) → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
87ex 413 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
9 simpl 483 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝐽 ∈ Top)
101eltopss 21443 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝑋)
111clsss3 21595 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝑋) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
1210, 11syldan 591 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
131ntrss2 21593 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
1412, 13syldan 591 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
151clsss 21590 . . . . . . . 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 1363 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)))
171clsidm 21603 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
1810, 17syldan 591 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
1916, 18sseqtrd 4004 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘𝑜))
201ntrss3 21596 . . . . . . . 8 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
2112, 20syldan 591 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
22 simpr 485 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝐽)
231sscls 21592 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝑋) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
2410, 23syldan 591 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
251ssntr 21594 . . . . . . . 8 (((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) ∧ (𝑜𝐽𝑜 ⊆ ((cls‘𝐽)‘𝑜))) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
269, 12, 22, 24, 25syl22anc 834 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
271clsss 21590 . . . . . . 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 1363 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
2919, 28eqssd 3981 . . . . 5 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
3029adantlr 711 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜))
31 2fveq3 6668 . . . . 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 2834 . . . 4 (𝐴 = ((cls‘𝐽)‘𝑜) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜)))
3430, 33syl5ibrcom 248 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
3534rexlimdva 3281 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
368, 35impbid 213 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 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  wss 3933   cuni 4830  cfv 6348  Topctop 21429  intcnt 21553  clsccl 21554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-cld 21555  df-ntr 21556  df-cls 21557
This theorem is referenced by: (None)
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