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Theorem flimclsi 19693
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 2454 . . . . . . . 8  |-  U. J  =  U. J
21flimfil 19684 . . . . . . 7  |-  ( x  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
32ad2antlr 726 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  ->  F  e.  ( Fil ` 
U. J ) )
4 flimnei 19682 . . . . . . 7  |-  ( ( x  e.  ( J 
fLim  F )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
y  e.  F )
54adantll 713 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
y  e.  F )
6 simpll 753 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  ->  S  e.  F )
7 filinn0 19575 . . . . . 6  |-  ( ( F  e.  ( Fil `  U. J )  /\  y  e.  F  /\  S  e.  F )  ->  ( y  i^i  S
)  =/=  (/) )
83, 5, 6, 7syl3anc 1219 . . . . 5  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
( y  i^i  S
)  =/=  (/) )
98ralrimiva 2830 . . . 4  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  A. y  e.  (
( nei `  J
) `  { x } ) ( y  i^i  S )  =/=  (/) )
10 flimtop 19680 . . . . . 6  |-  ( x  e.  ( J  fLim  F )  ->  J  e.  Top )
1110adantl 466 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  J  e.  Top )
12 filelss 19567 . . . . . . 7  |-  ( ( F  e.  ( Fil `  U. J )  /\  S  e.  F )  ->  S  C_  U. J )
1312ancoms 453 . . . . . 6  |-  ( ( S  e.  F  /\  F  e.  ( Fil ` 
U. J ) )  ->  S  C_  U. J
)
142, 13sylan2 474 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  S  C_  U. J )
151flimelbas 19683 . . . . . 6  |-  ( x  e.  ( J  fLim  F )  ->  x  e.  U. J )
1615adantl 466 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  x  e.  U. J )
171neindisj2 18869 . . . . 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 1219 . . . 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 232 . . 3  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  x  e.  ( ( cls `  J ) `  S ) )
2019ex 434 . 2  |-  ( S  e.  F  ->  (
x  e.  ( J 
fLim  F )  ->  x  e.  ( ( cls `  J
) `  S )
) )
2120ssrdv 3473 1  |-  ( S  e.  F  ->  ( J  fLim  F )  C_  ( ( cls `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    =/= wne 2648   A.wral 2799    i^i cin 3438    C_ wss 3439   (/)c0 3748   {csn 3988   U.cuni 4202   ` cfv 5529  (class class class)co 6203   Topctop 18640   clsccl 18764   neicnei 18843   Filcfil 19560    fLim cflim 19649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-fbas 17949  df-top 18645  df-cld 18765  df-ntr 18766  df-cls 18767  df-nei 18844  df-fil 19561  df-flim 19654
This theorem is referenced by:  flimcls  19700  flimfcls  19741  cnextcn  19781  cmetss  20967  minveclem4  21061
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