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Theorem fclsopn 23518
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 23517 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ )))
2 filn0 23366 . . . . . 6 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 β‰  βˆ…)
32adantl 483 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐹 β‰  βˆ…)
4 r19.2z 4495 . . . . . 6 ((𝐹 β‰  βˆ… ∧ βˆ€π‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ )) β†’ βˆƒπ‘  ∈ 𝐹 𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ))
54ex 414 . . . . 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 22415 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
87ad2antrr 725 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ 𝐽 ∈ Top)
9 filelss 23356 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑠 ∈ 𝐹) β†’ 𝑠 βŠ† 𝑋)
109adantll 713 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ 𝑠 βŠ† 𝑋)
11 toponuni 22416 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1211ad2antrr 725 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ 𝑋 = βˆͺ 𝐽)
1310, 12sseqtrd 4023 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ 𝑠 βŠ† βˆͺ 𝐽)
14 eqid 2733 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
1514clsss3 22563 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑠 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘ ) βŠ† βˆͺ 𝐽)
168, 13, 15syl2anc 585 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘ ) βŠ† βˆͺ 𝐽)
1716, 12sseqtrrd 4024 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘ ) βŠ† 𝑋)
1817sseld 3982 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝑠 ∈ 𝐹) β†’ (𝐴 ∈ ((clsβ€˜π½)β€˜π‘ ) β†’ 𝐴 ∈ 𝑋))
1918rexlimdva 3156 . . . 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 564 . 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 729 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝐽 ∈ Top)
2313adantlr 714 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝑠 βŠ† βˆͺ 𝐽)
24 simplr 768 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝐴 ∈ 𝑋)
2511ad3antrrr 729 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝑋 = βˆͺ 𝐽)
2624, 25eleqtrd 2836 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) β†’ 𝐴 ∈ βˆͺ 𝐽)
2714elcls 22577 . . . . . 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 1372 . . . . 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 3176 . . . 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 3287 . . . . 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 3180 . . . . . 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 3094 . . . . 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 275 . . . 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 287 . . 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 580 . 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 305 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  Topctop 22395  TopOnctopon 22412  clsccl 22522  Filcfil 23349   fClus cfcls 23440
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-nel 3048  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-ov 7412  df-oprab 7413  df-mpo 7414  df-fbas 20941  df-top 22396  df-topon 22413  df-cld 22523  df-ntr 22524  df-cls 22525  df-fil 23350  df-fcls 23445
This theorem is referenced by:  fclsopni  23519  fclselbas  23520  fclsnei  23523  fclsbas  23525  fclsss1  23526  fclsrest  23528  fclscf  23529  isfcf  23538
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