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Theorem fclsopn 23738
Description: Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
fclsopn ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))))
Distinct variable groups:   π‘œ,𝑠,𝐴   π‘œ,𝐹,𝑠   π‘œ,𝐽,𝑠   π‘œ,𝑋,𝑠

Proof of Theorem fclsopn
StepHypRef Expression
1 isfcls2 23737 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ )))
2 filn0 23586 . . . . . 6 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 β‰  βˆ…)
32adantl 482 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐹 β‰  βˆ…)
4 r19.2z 4494 . . . . . 6 ((𝐹 β‰  βˆ… ∧ βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ )) β†’ βˆƒπ‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ))
54ex 413 . . . . 5 (𝐹 β‰  βˆ… β†’ (βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) β†’ βˆƒπ‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ )))
63, 5syl 17 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) β†’ βˆƒπ‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ )))
7 topontop 22635 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
87ad2antrr 724 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ 𝐽 ∈ Top)
9 filelss 23576 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑠 ∈ 𝐹) β†’ 𝑠 βŠ† 𝑋)
109adantll 712 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ 𝑠 βŠ† 𝑋)
11 toponuni 22636 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1211ad2antrr 724 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ 𝑋 = βˆͺ 𝐽)
1310, 12sseqtrd 4022 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ 𝑠 βŠ† βˆͺ 𝐽)
14 eqid 2732 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
1514clsss3 22783 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑠 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘ ) βŠ† βˆͺ 𝐽)
168, 13, 15syl2anc 584 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘ ) βŠ† βˆͺ 𝐽)
1716, 12sseqtrrd 4023 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘ ) βŠ† 𝑋)
1817sseld 3981 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ (𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) β†’ 𝐴 ∈ 𝑋))
1918rexlimdva 3155 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (βˆƒπ‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) β†’ 𝐴 ∈ 𝑋))
206, 19syld 47 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) β†’ 𝐴 ∈ 𝑋))
2120pm4.71rd 563 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ))))
227ad3antrrr 728 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝐽 ∈ Top)
2313adantlr 713 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝑠 βŠ† βˆͺ 𝐽)
24 simplr 767 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝐴 ∈ 𝑋)
2511ad3antrrr 728 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝑋 = βˆͺ 𝐽)
2624, 25eleqtrd 2835 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝐴 ∈ βˆͺ 𝐽)
2714elcls 22797 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠 βŠ† βˆͺ 𝐽 ∧ 𝐴 ∈ βˆͺ 𝐽) β†’ (𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) ↔ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ (π‘œ ∩ 𝑠) β‰  βˆ…)))
2822, 23, 26, 27syl3anc 1371 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ (𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) ↔ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ (π‘œ ∩ 𝑠) β‰  βˆ…)))
2928ralbidva 3175 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) ↔ βˆ€π‘  ∈ 𝐹 βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ (π‘œ ∩ 𝑠) β‰  βˆ…)))
30 ralcom 3286 . . . . 5 (βˆ€π‘  ∈ 𝐹 βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ (π‘œ ∩ 𝑠) β‰  βˆ…) ↔ βˆ€π‘œ ∈ 𝐽 βˆ€π‘  ∈ 𝐹 (𝐴 ∈ π‘œ β†’ (π‘œ ∩ 𝑠) β‰  βˆ…))
31 r19.21v 3179 . . . . . 6 (βˆ€π‘  ∈ 𝐹 (𝐴 ∈ π‘œ β†’ (π‘œ ∩ 𝑠) β‰  βˆ…) ↔ (𝐴 ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))
3231ralbii 3093 . . . . 5 (βˆ€π‘œ ∈ 𝐽 βˆ€π‘  ∈ 𝐹 (𝐴 ∈ π‘œ β†’ (π‘œ ∩ 𝑠) β‰  βˆ…) ↔ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))
3330, 32bitri 274 . . . 4 (βˆ€π‘  ∈ 𝐹 βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ (π‘œ ∩ 𝑠) β‰  βˆ…) ↔ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))
3429, 33bitrdi 286 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) ↔ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…)))
3534pm5.32da 579 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ ((𝐴 ∈ 𝑋 ∧ βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ )) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))))
361, 21, 353bitrd 304 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (𝐴 ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7411  Topctop 22615  TopOnctopon 22632  clsccl 22742  Filcfil 23569   fClus cfcls 23660
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-nel 3047  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-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-top 22616  df-topon 22633  df-cld 22743  df-ntr 22744  df-cls 22745  df-fil 23570  df-fcls 23665
This theorem is referenced by:  fclsopni  23739  fclselbas  23740  fclsnei  23743  fclsbas  23745  fclsss1  23746  fclsrest  23748  fclscf  23749  isfcf  23758
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