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Theorem opnregcld 34855
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 22423 . . . 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 6846 . . . . 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 3599 . . . 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 22279 . . . . . . . . 9 ((𝐽 ∈ Top ∧ π‘œ ∈ 𝐽) β†’ π‘œ βŠ† 𝑋)
111clsss3 22433 . . . . . . . . 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 22431 . . . . . . . . 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 22428 . . . . . . . 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 22441 . . . . . . . 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 3988 . . . . . 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 22434 . . . . . . . 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 22430 . . . . . . . . 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 22432 . . . . . . . 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 22428 . . . . . . 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 3965 . . . . 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 6851 . . . . 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 247 . . 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 3149 . 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 3070   βŠ† wss 3914  βˆͺ cuni 4869  β€˜cfv 6500  Topctop 22265  intcnt 22391  clsccl 22392
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22266  df-cld 22393  df-ntr 22394  df-cls 22395
This theorem is referenced by: (None)
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