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Theorem flimclsi 18002
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 2435 . . . . . . . 8  |-  U. J  =  U. J
21flimfil 17993 . . . . . . 7  |-  ( x  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
32ad2antlr 708 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  ->  F  e.  ( Fil ` 
U. J ) )
4 flimnei 17991 . . . . . . 7  |-  ( ( x  e.  ( J 
fLim  F )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
y  e.  F )
54adantll 695 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
y  e.  F )
6 simpll 731 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  ->  S  e.  F )
7 filinn0 17884 . . . . . 6  |-  ( ( F  e.  ( Fil `  U. J )  /\  y  e.  F  /\  S  e.  F )  ->  ( y  i^i  S
)  =/=  (/) )
83, 5, 6, 7syl3anc 1184 . . . . 5  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
( y  i^i  S
)  =/=  (/) )
98ralrimiva 2781 . . . 4  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  A. y  e.  (
( nei `  J
) `  { x } ) ( y  i^i  S )  =/=  (/) )
10 flimtop 17989 . . . . . 6  |-  ( x  e.  ( J  fLim  F )  ->  J  e.  Top )
1110adantl 453 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  J  e.  Top )
12 filelss 17876 . . . . . . 7  |-  ( ( F  e.  ( Fil `  U. J )  /\  S  e.  F )  ->  S  C_  U. J )
1312ancoms 440 . . . . . 6  |-  ( ( S  e.  F  /\  F  e.  ( Fil ` 
U. J ) )  ->  S  C_  U. J
)
142, 13sylan2 461 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  S  C_  U. J )
151flimelbas 17992 . . . . . 6  |-  ( x  e.  ( J  fLim  F )  ->  x  e.  U. J )
1615adantl 453 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  x  e.  U. J )
171neindisj2 17179 . . . . 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 1184 . . . 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 224 . . 3  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  x  e.  ( ( cls `  J ) `  S ) )
2019ex 424 . 2  |-  ( S  e.  F  ->  (
x  e.  ( J 
fLim  F )  ->  x  e.  ( ( cls `  J
) `  S )
) )
2120ssrdv 3346 1  |-  ( S  e.  F  ->  ( J  fLim  F )  C_  ( ( cls `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    =/= wne 2598   A.wral 2697    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   U.cuni 4007   ` cfv 5446  (class class class)co 6073   Topctop 16950   clsccl 17074   neicnei 17153   Filcfil 17869    fLim cflim 17958
This theorem is referenced by:  flimcls  18009  flimfcls  18050  cnextcn  18090  cmetss  19259  minveclem4  19325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-fbas 16691  df-top 16955  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-fil 17870  df-flim 17963
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