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Theorem cldregopn 36313
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 23070 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽))
3 eqcom 2741 . . . . 5 (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
43biimpi 216 . . . 4 (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
5 fveq2 6906 . . . . 5 (𝑐 = ((cls‘𝐽)‘𝐴) → ((int‘𝐽)‘𝑐) = ((int‘𝐽)‘((cls‘𝐽)‘𝐴)))
65rspceeqv 3644 . . . 4 ((((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
72, 4, 6syl2an 596 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))
87ex 412 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)))
9 cldrcl 23049 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
101cldss 23052 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → 𝑐𝑋)
111ntrss2 23080 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
129, 10, 11syl2anc 584 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑐)
131clsss2 23095 . . . . . . . 8 ((𝑐 ∈ (Clsd‘𝐽) ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑐) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
1412, 13mpdan 687 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐)
151ntrss 23078 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐𝑋 ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
169, 10, 14, 15syl3anc 1370 . . . . . 6 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐))
171ntridm 23091 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
189, 10, 17syl2anc 584 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐))
191ntrss3 23083 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑐𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
209, 10, 19syl2anc 584 . . . . . . . . 9 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑋)
211clsss3 23082 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
229, 20, 21syl2anc 584 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋)
231sscls 23079 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
249, 20, 23syl2anc 584 . . . . . . . 8 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)))
251ntrss 23078 . . . . . . . 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 1370 . . . . . . 7 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
2718, 26eqsstrrd 4034 . . . . . 6 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))))
2816, 27eqssd 4012 . . . . 5 (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
2928adantl 481 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))
30 2fveq3 6911 . . . . 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 2750 . . . 4 (𝐴 = ((int‘𝐽)‘𝑐) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐)))
3329, 32syl5ibrcom 247 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
3433rexlimdva 3152 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴))
358, 34impbid 212 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 206  wa 395   = wceq 1536  wcel 2105  wrex 3067  wss 3962   cuni 4911  cfv 6562  Topctop 22914  Clsdccld 23039  intcnt 23040  clsccl 23041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-cld 23042  df-ntr 23043  df-cls 23044
This theorem is referenced by: (None)
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