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Theorem flimclsi 17673
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 2283 . . . . . . . 8  |-  U. J  =  U. J
21flimfil 17664 . . . . . . 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 17662 . . . . . . 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 17555 . . . . . 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 2626 . . . 4  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  A. y  e.  (
( nei `  J
) `  { x } ) ( y  i^i  S )  =/=  (/) )
10 flimtop 17660 . . . . . 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 17547 . . . . . . 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 17663 . . . . . 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 16860 . . . . 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 3185 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 1684    =/= wne 2446   A.wral 2543    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   ` cfv 5255  (class class class)co 5858   Topctop 16631   clsccl 16755   neicnei 16834   Filcfil 17540    fLim cflim 17629
This theorem is referenced by:  flimcls  17680  flimfcls  17721  cmetss  18740  minveclem4  18796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-fbas 17520  df-fil 17541  df-flim 17634
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