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Theorem cldregopn 35815
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 = βˆͺ 𝐽
Assertion
Ref Expression
cldregopn ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴 ↔ βˆƒπ‘ ∈ (Clsdβ€˜π½)𝐴 = ((intβ€˜π½)β€˜π‘)))
Distinct variable groups:   𝐴,𝑐   𝐽,𝑐   𝑋,𝑐

Proof of Theorem cldregopn
StepHypRef Expression
1 opnregcld.1 . . . . 5 𝑋 = βˆͺ 𝐽
21clscld 22964 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π΄) ∈ (Clsdβ€˜π½))
3 eqcom 2735 . . . . 5 (((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴 ↔ 𝐴 = ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)))
43biimpi 215 . . . 4 (((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴 β†’ 𝐴 = ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)))
5 fveq2 6897 . . . . 5 (𝑐 = ((clsβ€˜π½)β€˜π΄) β†’ ((intβ€˜π½)β€˜π‘) = ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)))
65rspceeqv 3631 . . . 4 ((((clsβ€˜π½)β€˜π΄) ∈ (Clsdβ€˜π½) ∧ 𝐴 = ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜π½)𝐴 = ((intβ€˜π½)β€˜π‘))
72, 4, 6syl2an 595 . . 3 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴) β†’ βˆƒπ‘ ∈ (Clsdβ€˜π½)𝐴 = ((intβ€˜π½)β€˜π‘))
87ex 412 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴 β†’ βˆƒπ‘ ∈ (Clsdβ€˜π½)𝐴 = ((intβ€˜π½)β€˜π‘)))
9 cldrcl 22943 . . . . . . 7 (𝑐 ∈ (Clsdβ€˜π½) β†’ 𝐽 ∈ Top)
101cldss 22946 . . . . . . 7 (𝑐 ∈ (Clsdβ€˜π½) β†’ 𝑐 βŠ† 𝑋)
111ntrss2 22974 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑐 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑐)
129, 10, 11syl2anc 583 . . . . . . . 8 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑐)
131clsss2 22989 . . . . . . . 8 ((𝑐 ∈ (Clsdβ€˜π½) ∧ ((intβ€˜π½)β€˜π‘) βŠ† 𝑐) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) βŠ† 𝑐)
1412, 13mpdan 686 . . . . . . 7 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) βŠ† 𝑐)
151ntrss 22972 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑐 βŠ† 𝑋 ∧ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) βŠ† 𝑐) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))) βŠ† ((intβ€˜π½)β€˜π‘))
169, 10, 14, 15syl3anc 1369 . . . . . 6 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))) βŠ† ((intβ€˜π½)β€˜π‘))
171ntridm 22985 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑐 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) = ((intβ€˜π½)β€˜π‘))
189, 10, 17syl2anc 583 . . . . . . 7 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((intβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) = ((intβ€˜π½)β€˜π‘))
191ntrss3 22977 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑐 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑋)
209, 10, 19syl2anc 583 . . . . . . . . 9 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((intβ€˜π½)β€˜π‘) βŠ† 𝑋)
211clsss3 22976 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((intβ€˜π½)β€˜π‘) βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) βŠ† 𝑋)
229, 20, 21syl2anc 583 . . . . . . . 8 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) βŠ† 𝑋)
231sscls 22973 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((intβ€˜π½)β€˜π‘) βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘) βŠ† ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)))
249, 20, 23syl2anc 583 . . . . . . . 8 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((intβ€˜π½)β€˜π‘) βŠ† ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)))
251ntrss 22972 . . . . . . . 8 ((𝐽 ∈ Top ∧ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) βŠ† 𝑋 ∧ ((intβ€˜π½)β€˜π‘) βŠ† ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))) β†’ ((intβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))))
269, 22, 24, 25syl3anc 1369 . . . . . . 7 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((intβ€˜π½)β€˜((intβ€˜π½)β€˜π‘)) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))))
2718, 26eqsstrrd 4019 . . . . . 6 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((intβ€˜π½)β€˜π‘) βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))))
2816, 27eqssd 3997 . . . . 5 (𝑐 ∈ (Clsdβ€˜π½) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))) = ((intβ€˜π½)β€˜π‘))
2928adantl 481 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ 𝑐 ∈ (Clsdβ€˜π½)) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))) = ((intβ€˜π½)β€˜π‘))
30 2fveq3 6902 . . . . 5 (𝐴 = ((intβ€˜π½)β€˜π‘) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))))
31 id 22 . . . . 5 (𝐴 = ((intβ€˜π½)β€˜π‘) β†’ 𝐴 = ((intβ€˜π½)β€˜π‘))
3230, 31eqeq12d 2744 . . . 4 (𝐴 = ((intβ€˜π½)β€˜π‘) β†’ (((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴 ↔ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π‘))) = ((intβ€˜π½)β€˜π‘)))
3329, 32syl5ibrcom 246 . . 3 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ 𝑐 ∈ (Clsdβ€˜π½)) β†’ (𝐴 = ((intβ€˜π½)β€˜π‘) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴))
3433rexlimdva 3152 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (βˆƒπ‘ ∈ (Clsdβ€˜π½)𝐴 = ((intβ€˜π½)β€˜π‘) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴))
358, 34impbid 211 1 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴 ↔ βˆƒπ‘ ∈ (Clsdβ€˜π½)𝐴 = ((intβ€˜π½)β€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3067   βŠ† wss 3947  βˆͺ cuni 4908  β€˜cfv 6548  Topctop 22808  Clsdccld 22933  intcnt 22934  clsccl 22935
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 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-top 22809  df-cld 22936  df-ntr 22937  df-cls 22938
This theorem is referenced by: (None)
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