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Theorem flimsncls 20936
Description: If  A is a limit point of the filter  F, then all the points which specialize  A (in the specialization preorder) are also limit points. Thus, the set of limit points is a union of closed sets (although this is only nontrivial for non-T1 spaces). (Contributed by Mario Carneiro, 20-Sep-2015.)
Assertion
Ref Expression
flimsncls  |-  ( A  e.  ( J  fLim  F )  ->  ( ( cls `  J ) `  { A } )  C_  ( J  fLim  F ) )

Proof of Theorem flimsncls
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimtop 20915 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
2 eqid 2429 . . . . . . . 8  |-  U. J  =  U. J
32flimelbas 20918 . . . . . . 7  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  U. J )
43snssd 4148 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  { A }  C_  U. J )
52clsss3 20009 . . . . . 6  |-  ( ( J  e.  Top  /\  { A }  C_  U. J
)  ->  ( ( cls `  J ) `  { A } )  C_  U. J )
61, 4, 5syl2anc 665 . . . . 5  |-  ( A  e.  ( J  fLim  F )  ->  ( ( cls `  J ) `  { A } )  C_  U. J )
76sselda 3470 . . . 4  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  ->  x  e.  U. J )
8 simpll 758 . . . . . . 7  |-  ( ( ( A  e.  ( J  fLim  F )  /\  x  e.  (
( cls `  J
) `  { A } ) )  /\  ( y  e.  J  /\  x  e.  y
) )  ->  A  e.  ( J  fLim  F
) )
98, 1syl 17 . . . . . . . 8  |-  ( ( ( A  e.  ( J  fLim  F )  /\  x  e.  (
( cls `  J
) `  { A } ) )  /\  ( y  e.  J  /\  x  e.  y
) )  ->  J  e.  Top )
10 simprl 762 . . . . . . . 8  |-  ( ( ( A  e.  ( J  fLim  F )  /\  x  e.  (
( cls `  J
) `  { A } ) )  /\  ( y  e.  J  /\  x  e.  y
) )  ->  y  e.  J )
111adantr 466 . . . . . . . . . 10  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  ->  J  e.  Top )
124adantr 466 . . . . . . . . . 10  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  ->  { A }  C_  U. J
)
13 simpr 462 . . . . . . . . . 10  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  ->  x  e.  ( ( cls `  J ) `  { A } ) )
1411, 12, 133jca 1185 . . . . . . . . 9  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  -> 
( J  e.  Top  /\ 
{ A }  C_  U. J  /\  x  e.  ( ( cls `  J
) `  { A } ) ) )
152clsndisj 20026 . . . . . . . . . 10  |-  ( ( ( J  e.  Top  /\ 
{ A }  C_  U. J  /\  x  e.  ( ( cls `  J
) `  { A } ) )  /\  ( y  e.  J  /\  x  e.  y
) )  ->  (
y  i^i  { A } )  =/=  (/) )
16 disjsn 4063 . . . . . . . . . . 11  |-  ( ( y  i^i  { A } )  =  (/)  <->  -.  A  e.  y )
1716necon2abii 2697 . . . . . . . . . 10  |-  ( A  e.  y  <->  ( y  i^i  { A } )  =/=  (/) )
1815, 17sylibr 215 . . . . . . . . 9  |-  ( ( ( J  e.  Top  /\ 
{ A }  C_  U. J  /\  x  e.  ( ( cls `  J
) `  { A } ) )  /\  ( y  e.  J  /\  x  e.  y
) )  ->  A  e.  y )
1914, 18sylan 473 . . . . . . . 8  |-  ( ( ( A  e.  ( J  fLim  F )  /\  x  e.  (
( cls `  J
) `  { A } ) )  /\  ( y  e.  J  /\  x  e.  y
) )  ->  A  e.  y )
20 opnneip 20070 . . . . . . . 8  |-  ( ( J  e.  Top  /\  y  e.  J  /\  A  e.  y )  ->  y  e.  ( ( nei `  J ) `
 { A }
) )
219, 10, 19, 20syl3anc 1264 . . . . . . 7  |-  ( ( ( A  e.  ( J  fLim  F )  /\  x  e.  (
( cls `  J
) `  { A } ) )  /\  ( y  e.  J  /\  x  e.  y
) )  ->  y  e.  ( ( nei `  J
) `  { A } ) )
22 flimnei 20917 . . . . . . 7  |-  ( ( A  e.  ( J 
fLim  F )  /\  y  e.  ( ( nei `  J
) `  { A } ) )  -> 
y  e.  F )
238, 21, 22syl2anc 665 . . . . . 6  |-  ( ( ( A  e.  ( J  fLim  F )  /\  x  e.  (
( cls `  J
) `  { A } ) )  /\  ( y  e.  J  /\  x  e.  y
) )  ->  y  e.  F )
2423expr 618 . . . . 5  |-  ( ( ( A  e.  ( J  fLim  F )  /\  x  e.  (
( cls `  J
) `  { A } ) )  /\  y  e.  J )  ->  ( x  e.  y  ->  y  e.  F
) )
2524ralrimiva 2846 . . . 4  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  ->  A. y  e.  J  ( x  e.  y  ->  y  e.  F ) )
262toptopon 19883 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
2711, 26sylib 199 . . . . 5  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  ->  J  e.  (TopOn `  U. J ) )
282flimfil 20919 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
2928adantr 466 . . . . 5  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  ->  F  e.  ( Fil ` 
U. J ) )
30 flimopn 20925 . . . . 5  |-  ( ( J  e.  (TopOn `  U. J )  /\  F  e.  ( Fil `  U. J ) )  -> 
( x  e.  ( J  fLim  F )  <->  ( x  e.  U. J  /\  A. y  e.  J  ( x  e.  y  ->  y  e.  F ) ) ) )
3127, 29, 30syl2anc 665 . . . 4  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  -> 
( x  e.  ( J  fLim  F )  <->  ( x  e.  U. J  /\  A. y  e.  J  ( x  e.  y  ->  y  e.  F ) ) ) )
327, 25, 31mpbir2and 930 . . 3  |-  ( ( A  e.  ( J 
fLim  F )  /\  x  e.  ( ( cls `  J
) `  { A } ) )  ->  x  e.  ( J  fLim  F ) )
3332ex 435 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( x  e.  ( ( cls `  J
) `  { A } )  ->  x  e.  ( J  fLim  F
) ) )
3433ssrdv 3476 1  |-  ( A  e.  ( J  fLim  F )  ->  ( ( cls `  J ) `  { A } )  C_  ( J  fLim  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1870    =/= wne 2625   A.wral 2782    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002   U.cuni 4222   ` cfv 5601  (class class class)co 6305   Topctop 19852  TopOnctopon 19853   clsccl 19968   neicnei 20048   Filcfil 20795    fLim cflim 20884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-fbas 18906  df-top 19856  df-topon 19858  df-cld 19969  df-ntr 19970  df-cls 19971  df-nei 20049  df-fil 20796  df-flim 20889
This theorem is referenced by:  tsmscls  21087
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