Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opnregcld Structured version   Visualization version   GIF version
Theorem opnregcld 35215
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 = βˆͺ 𝐽
Assertion
Ref Expression
opnregcld ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴 ↔ βˆƒπ‘œ ∈ 𝐽 𝐴 = ((clsβ€˜π½)β€˜π‘œ)))
Distinct variable groups:   𝐴,π‘œ   π‘œ,𝐽   π‘œ,𝑋
Proof of Theorem opnregcld
StepHypRef Expression
1 opnregcld.1 . . . . 5 𝑋 = βˆͺ 𝐽
21ntropn 22553 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) ∈ 𝐽)
3 eqcom 2740 . . . . 5 (((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴 ↔ 𝐴 = ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)))
43biimpi 215 . . . 4 (((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴 β†’ 𝐴 = ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)))
5 fveq2 6892 . . . . 5 (π‘œ = ((intβ€˜π½)β€˜π΄) β†’ ((clsβ€˜π½)β€˜π‘œ) = ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)))
65rspceeqv 3634 . . . 4 ((((intβ€˜π½)β€˜π΄) ∈ 𝐽 ∧ 𝐴 = ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄))) β†’ βˆƒπ‘œ ∈ 𝐽 𝐴 = ((clsβ€˜π½)β€˜π‘œ))
72, 4, 6syl2an 597 . . 3 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴) β†’ βˆƒπ‘œ ∈ 𝐽 𝐴 = ((clsβ€˜π½)β€˜π‘œ))
87ex 414 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴 β†’ βˆƒπ‘œ ∈ 𝐽 𝐴 = ((clsβ€˜π½)β€˜π‘œ)))
9 simpl 484 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ 𝐽 ∈ Top)
101eltopss 22409 . . . . . . . . 9 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ π‘œ βŠ† 𝑋)
111clsss3 22563 . . . . . . . . 9 ((𝐽 ∈ Top ∧ π‘œ βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘œ) βŠ† 𝑋)
1210, 11syldan 592 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘œ) βŠ† 𝑋)
131ntrss2 22561 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((clsβ€˜π½)β€˜π‘œ) βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)) βŠ† ((clsβ€˜π½)β€˜π‘œ))
1412, 13syldan 592 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)) βŠ† ((clsβ€˜π½)β€˜π‘œ))
151clsss 22558 . . . . . . . 8 ((𝐽 ∈ Top ∧ ((clsβ€˜π½)β€˜π‘œ) βŠ† 𝑋 ∧ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)) βŠ† ((clsβ€˜π½)β€˜π‘œ)) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))) βŠ† ((clsβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)))
169, 12, 14, 15syl3anc 1372 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))) βŠ† ((clsβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)))
171clsidm 22571 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)) = ((clsβ€˜π½)β€˜π‘œ))
1810, 17syldan 592 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((clsβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)) = ((clsβ€˜π½)β€˜π‘œ))
1916, 18sseqtrd 4023 . . . . . 6 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))) βŠ† ((clsβ€˜π½)β€˜π‘œ))
201ntrss3 22564 . . . . . . . 8 ((𝐽 ∈ Top ∧ ((clsβ€˜π½)β€˜π‘œ) βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)) βŠ† 𝑋)
2112, 20syldan 592 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)) βŠ† 𝑋)
22 simpr 486 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ π‘œ ∈ 𝐽)
231sscls 22560 . . . . . . . . 9 ((𝐽 ∈ Top ∧ π‘œ βŠ† 𝑋) β†’ π‘œ βŠ† ((clsβ€˜π½)β€˜π‘œ))
2410, 23syldan 592 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ π‘œ βŠ† ((clsβ€˜π½)β€˜π‘œ))
251ssntr 22562 . . . . . . . 8 (((𝐽 ∈ Top ∧ ((clsβ€˜π½)β€˜π‘œ) βŠ† 𝑋) ∧ (π‘œ ∈ 𝐽 ∧ π‘œ βŠ† ((clsβ€˜π½)β€˜π‘œ))) β†’ π‘œ βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)))
269, 12, 22, 24, 25syl22anc 838 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ π‘œ βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)))
271clsss 22558 . . . . . . 7 ((𝐽 ∈ Top ∧ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)) βŠ† 𝑋 ∧ π‘œ βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))) β†’ ((clsβ€˜π½)β€˜π‘œ) βŠ† ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))))
289, 21, 26, 27syl3anc 1372 . . . . . 6 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘œ) βŠ† ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))))
2919, 28eqssd 4000 . . . . 5 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))) = ((clsβ€˜π½)β€˜π‘œ))
3029adantlr 714 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ π‘œ ∈ 𝐽) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))) = ((clsβ€˜π½)β€˜π‘œ))
31 2fveq3 6897 . . . . 5 (𝐴 = ((clsβ€˜π½)β€˜π‘œ) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))))
32 id 22 . . . . 5 (𝐴 = ((clsβ€˜π½)β€˜π‘œ) β†’ 𝐴 = ((clsβ€˜π½)β€˜π‘œ))
3331, 32eqeq12d 2749 . . . 4 (𝐴 = ((clsβ€˜π½)β€˜π‘œ) β†’ (((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴 ↔ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ))) = ((clsβ€˜π½)β€˜π‘œ)))
3430, 33syl5ibrcom 246 . . 3 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ π‘œ ∈ 𝐽) β†’ (𝐴 = ((clsβ€˜π½)β€˜π‘œ) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴))
3534rexlimdva 3156 . 2 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (βˆƒπ‘œ ∈ 𝐽 𝐴 = ((clsβ€˜π½)β€˜π‘œ) β†’ ((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴))
368, 35impbid 211 1 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴 ↔ βˆƒπ‘œ ∈ 𝐽 𝐴 = ((clsβ€˜π½)β€˜π‘œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  intcnt 22521  clsccl 22522
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22396  df-cld 22523  df-ntr 22524  df-cls 22525
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator