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Theorem opnregcld 35153
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 22535 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
3 eqcom 2740 . . . . 5 (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
43biimpi 215 . . . 4 (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
5 fveq2 6888 . . . . 5 (𝑜 = ((int‘𝐽)‘𝐴) → ((cls‘𝐽)‘𝑜) = ((cls‘𝐽)‘((int‘𝐽)‘𝐴)))
65rspceeqv 3632 . . . 4 ((((int‘𝐽)‘𝐴) ∈ 𝐽𝐴 = ((cls‘𝐽)‘((int‘𝐽)‘𝐴))) → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
72, 4, 6syl2an 597 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴) → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜))
87ex 414 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 → ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))
9 simpl 484 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝐽 ∈ Top)
101eltopss 22391 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝑋)
111clsss3 22545 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝑋) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
1210, 11syldan 592 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝑋)
131ntrss2 22543 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
1412, 13syldan 592 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ ((cls‘𝐽)‘𝑜))
151clsss 22540 . . . . . . . 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 1372 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)))
171clsidm 22553 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
1810, 17syldan 592 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑜)) = ((cls‘𝐽)‘𝑜))
1916, 18sseqtrd 4021 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) ⊆ ((cls‘𝐽)‘𝑜))
201ntrss3 22546 . . . . . . . 8 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
2112, 20syldan 592 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘𝑜)) ⊆ 𝑋)
22 simpr 486 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝐽)
231sscls 22542 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑜𝑋) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
2410, 23syldan 592 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜 ⊆ ((cls‘𝐽)‘𝑜))
251ssntr 22544 . . . . . . . 8 (((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑋) ∧ (𝑜𝐽𝑜 ⊆ ((cls‘𝐽)‘𝑜))) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
269, 12, 22, 24, 25syl22anc 838 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜 ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑜)))
271clsss 22540 . . . . . . 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 1372 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))))
2919, 28eqssd 3998 . . . . 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 6893 . . . . 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 2749 . . . 4 (𝐴 = ((cls‘𝐽)‘𝑜) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ((cls‘𝐽)‘((int‘𝐽)‘((cls‘𝐽)‘𝑜))) = ((cls‘𝐽)‘𝑜)))
3430, 33syl5ibrcom 246 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑜𝐽) → (𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
3534rexlimdva 3156 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜) → ((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴))
368, 35impbid 211 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 205  wa 397   = wceq 1542  wcel 2107  wrex 3071  wss 3947   cuni 4907  cfv 6540  Topctop 22377  intcnt 22503  clsccl 22504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 22378  df-cld 22505  df-ntr 22506  df-cls 22507
This theorem is referenced by: (None)
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