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Theorem fclssscls 17729
Description: The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclssscls  |-  ( S  e.  F  ->  ( J  fClus  F )  C_  ( ( cls `  J
) `  S )
)

Proof of Theorem fclssscls
Dummy variables  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . 5  |-  U. J  =  U. J
21isfcls 17720 . . . 4  |-  ( x  e.  ( J  fClus  F )  <->  ( J  e. 
Top  /\  F  e.  ( Fil `  U. J
)  /\  A. s  e.  F  x  e.  ( ( cls `  J
) `  s )
) )
32simp3bi 972 . . 3  |-  ( x  e.  ( J  fClus  F )  ->  A. s  e.  F  x  e.  ( ( cls `  J
) `  s )
)
4 fveq2 5541 . . . . 5  |-  ( s  =  S  ->  (
( cls `  J
) `  s )  =  ( ( cls `  J ) `  S
) )
54eleq2d 2363 . . . 4  |-  ( s  =  S  ->  (
x  e.  ( ( cls `  J ) `
 s )  <->  x  e.  ( ( cls `  J
) `  S )
) )
65rspcv 2893 . . 3  |-  ( S  e.  F  ->  ( A. s  e.  F  x  e.  ( ( cls `  J ) `  s )  ->  x  e.  ( ( cls `  J
) `  S )
) )
73, 6syl5 28 . 2  |-  ( S  e.  F  ->  (
x  e.  ( J 
fClus  F )  ->  x  e.  ( ( cls `  J
) `  S )
) )
87ssrdv 3198 1  |-  ( S  e.  F  ->  ( J  fClus  F )  C_  ( ( cls `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   U.cuni 3843   ` cfv 5271  (class class class)co 5874   Topctop 16647   clsccl 16771   Filcfil 17556    fClus cfcls 17647
This theorem is referenced by:  fclscmp  17741
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-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-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-fbas 17536  df-fil 17557  df-fcls 17652
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