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Theorem fclsss2 17770
Description: A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
fclsss2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( J  fClus  G )  C_  ( J  fClus  F ) )

Proof of Theorem fclsss2
Dummy variables  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 960 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  F  C_  G
)
2 ssralv 3271 . . . . . 6  |-  ( F 
C_  G  ->  ( A. s  e.  G  x  e.  ( ( cls `  J ) `  s )  ->  A. s  e.  F  x  e.  ( ( cls `  J
) `  s )
) )
31, 2syl 15 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  ( A. s  e.  G  x  e.  ( ( cls `  J
) `  s )  ->  A. s  e.  F  x  e.  ( ( cls `  J ) `  s ) ) )
4 simpl1 958 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  J  e.  (TopOn `  X ) )
5 fclstopon 17759 . . . . . . . 8  |-  ( x  e.  ( J  fClus  G )  ->  ( J  e.  (TopOn `  X )  <->  G  e.  ( Fil `  X
) ) )
65adantl 452 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  ( J  e.  (TopOn `  X )  <->  G  e.  ( Fil `  X
) ) )
74, 6mpbid 201 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  G  e.  ( Fil `  X ) )
8 isfcls2 17760 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  G  e.  ( Fil `  X
) )  ->  (
x  e.  ( J 
fClus  G )  <->  A. s  e.  G  x  e.  ( ( cls `  J
) `  s )
) )
94, 7, 8syl2anc 642 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  ( x  e.  ( J  fClus  G )  <->  A. s  e.  G  x  e.  ( ( cls `  J ) `  s ) ) )
10 simpl2 959 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  F  e.  ( Fil `  X ) )
11 isfcls2 17760 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  ( J 
fClus  F )  <->  A. s  e.  F  x  e.  ( ( cls `  J
) `  s )
) )
124, 10, 11syl2anc 642 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  ( x  e.  ( J  fClus  F )  <->  A. s  e.  F  x  e.  ( ( cls `  J ) `  s ) ) )
133, 9, 123imtr4d 259 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  ( x  e.  ( J  fClus  G )  ->  x  e.  ( J  fClus  F )
) )
1413ex 423 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  ( J  fClus  G )  ->  ( x  e.  ( J  fClus  G )  ->  x  e.  ( J  fClus  F )
) ) )
1514pm2.43d 44 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  ( J  fClus  G )  ->  x  e.  ( J  fClus  F )
) )
1615ssrdv 3219 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( J  fClus  G )  C_  ( J  fClus  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1701   A.wral 2577    C_ wss 3186   ` cfv 5292  (class class class)co 5900  TopOnctopon 16688   clsccl 16811   Filcfil 17592    fClus cfcls 17683
This theorem is referenced by:  fclsfnflim  17774  ufilcmp  17779  cnpfcfi  17787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-fbas 16429  df-topon 16695  df-fil 17593  df-fcls 17688
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