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Theorem opnregcld 35301
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 22560 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π΄) ∈ 𝐽)
3 eqcom 2739 . . . . 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 6891 . . . . 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 3633 . . . 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 596 . . 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 413 . 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 483 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ 𝐽 ∈ Top)
101eltopss 22416 . . . . . . . . 9 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ π‘œ βŠ† 𝑋)
111clsss3 22570 . . . . . . . . 9 ((𝐽 ∈ Top ∧ π‘œ βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘œ) βŠ† 𝑋)
1210, 11syldan 591 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘œ) βŠ† 𝑋)
131ntrss2 22568 . . . . . . . . 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 591 . . . . . . . 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 22565 . . . . . . . 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 1371 . . . . . . 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 22578 . . . . . . . 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 591 . . . . . . 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 4022 . . . . . 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 22571 . . . . . . . 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 591 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)) βŠ† 𝑋)
22 simpr 485 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ π‘œ ∈ 𝐽)
231sscls 22567 . . . . . . . . 9 ((𝐽 ∈ Top ∧ π‘œ βŠ† 𝑋) β†’ π‘œ βŠ† ((clsβ€˜π½)β€˜π‘œ))
2410, 23syldan 591 . . . . . . . 8 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ π‘œ βŠ† ((clsβ€˜π½)β€˜π‘œ))
251ssntr 22569 . . . . . . . 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 837 . . . . . . 7 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ π‘œ βŠ† ((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π‘œ)))
271clsss 22565 . . . . . . 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 1371 . . . . . 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 3999 . . . . 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 713 . . . 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 6896 . . . . 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 2748 . . . 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 3155 . 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 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22402  intcnt 22528  clsccl 22529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 22403  df-cld 22530  df-ntr 22531  df-cls 22532
This theorem is referenced by: (None)
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