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Theorem flimclsi 17689
Description: The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
flimclsi  |-  ( S  e.  F  ->  ( J  fLim  F )  C_  ( ( cls `  J
) `  S )
)

Proof of Theorem flimclsi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . . . . 8  |-  U. J  =  U. J
21flimfil 17680 . . . . . . 7  |-  ( x  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
32ad2antlr 707 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  ->  F  e.  ( Fil ` 
U. J ) )
4 flimnei 17678 . . . . . . 7  |-  ( ( x  e.  ( J 
fLim  F )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
y  e.  F )
54adantll 694 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
y  e.  F )
6 simpll 730 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  ->  S  e.  F )
7 filinn0 17571 . . . . . 6  |-  ( ( F  e.  ( Fil `  U. J )  /\  y  e.  F  /\  S  e.  F )  ->  ( y  i^i  S
)  =/=  (/) )
83, 5, 6, 7syl3anc 1182 . . . . 5  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
( y  i^i  S
)  =/=  (/) )
98ralrimiva 2639 . . . 4  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  A. y  e.  (
( nei `  J
) `  { x } ) ( y  i^i  S )  =/=  (/) )
10 flimtop 17676 . . . . . 6  |-  ( x  e.  ( J  fLim  F )  ->  J  e.  Top )
1110adantl 452 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  J  e.  Top )
12 filelss 17563 . . . . . . 7  |-  ( ( F  e.  ( Fil `  U. J )  /\  S  e.  F )  ->  S  C_  U. J )
1312ancoms 439 . . . . . 6  |-  ( ( S  e.  F  /\  F  e.  ( Fil ` 
U. J ) )  ->  S  C_  U. J
)
142, 13sylan2 460 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  S  C_  U. J )
151flimelbas 17679 . . . . . 6  |-  ( x  e.  ( J  fLim  F )  ->  x  e.  U. J )
1615adantl 452 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  x  e.  U. J )
171neindisj2 16876 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J  /\  x  e.  U. J )  ->  ( x  e.  ( ( cls `  J
) `  S )  <->  A. y  e.  ( ( nei `  J ) `
 { x }
) ( y  i^i 
S )  =/=  (/) ) )
1811, 14, 16, 17syl3anc 1182 . . . 4  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  -> 
( x  e.  ( ( cls `  J
) `  S )  <->  A. y  e.  ( ( nei `  J ) `
 { x }
) ( y  i^i 
S )  =/=  (/) ) )
199, 18mpbird 223 . . 3  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  x  e.  ( ( cls `  J ) `  S ) )
2019ex 423 . 2  |-  ( S  e.  F  ->  (
x  e.  ( J 
fLim  F )  ->  x  e.  ( ( cls `  J
) `  S )
) )
2120ssrdv 3198 1  |-  ( S  e.  F  ->  ( J  fLim  F )  C_  ( ( cls `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696    =/= wne 2459   A.wral 2556    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   ` cfv 5271  (class class class)co 5874   Topctop 16647   clsccl 16771   neicnei 16850   Filcfil 17556    fLim cflim 17645
This theorem is referenced by:  flimcls  17696  flimfcls  17737  cmetss  18756  minveclem4  18812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-top 16652  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-fbas 17536  df-fil 17557  df-flim 17650
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