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Theorem cldregopn 36053
Description: A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)
Hypothesis
Ref Expression
opnregcld.1 𝑋 = 𝐽
Assertion
Ref Expression
cldregopn ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
Distinct variable groups:   𝐴,𝑐   𝐽,𝑐   𝑋,𝑐

Proof of Theorem cldregopn
StepHypRef Expression
1 opnregcld.1 . . . . 5 𝑋 = 𝐽
21clscld 23036 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽))
3 eqcom 2733 . . . . 5 (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
43biimpi 215 . . . 4 (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
5 fveq2 6890 . . . . 5 (𝑐 = ((cls‘𝐽)‘𝐴) → ((int‘𝐽)‘𝑐) = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
65rspceeqv 3629 . . . 4 ((((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
72, 4, 6syl2an 594 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
87ex 411 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
9 cldrcl 23015 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
101cldss 23018 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → 𝑐𝑋)
111ntrss2 23046 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
129, 10, 11syl2anc 582 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
131clsss2 23061 . . . . . . . 8 ((𝑐 ∈ (Clsd‘𝐽) ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑐) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
1412, 13mpdan 685 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
151ntrss 23044 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐𝑋 ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
169, 10, 14, 15syl3anc 1368 . . . . . 6 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
171ntridm 23057 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
189, 10, 17syl2anc 582 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
191ntrss3 23049 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
209, 10, 19syl2anc 582 . . . . . . . . 9 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
211clsss3 23048 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
229, 20, 21syl2anc 582 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
231sscls 23045 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
249, 20, 23syl2anc 582 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
251ntrss 23044 . . . . . . . 8 ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐))) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
269, 22, 24, 25syl3anc 1368 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
2718, 26eqsstrrd 4018 . . . . . 6 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
2816, 27eqssd 3996 . . . . 5 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
2928adantl 480 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
30 2fveq3 6895 . . . . 5 (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
31 id 22 . . . . 5 (𝐴 = ((int‘𝐽)‘𝑐) → 𝐴 = ((int‘𝐽)‘𝑐))
3230, 31eqeq12d 2742 . . . 4 (𝐴 = ((int‘𝐽)‘𝑐) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐)))
3329, 32syl5ibrcom 246 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
3433rexlimdva 3145 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
358, 34impbid 211 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wrex 3060  wss 3946   cuni 4905  cfv 6543  Topctop 22880  Clsdccld 23005  intcnt 23006  clsccl 23007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22881  df-cld 23008  df-ntr 23009  df-cls 23010
This theorem is referenced by: (None)
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