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Theorem fclsss2 20369
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 1001 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  F  C_  G
)
2 ssralv 3569 . . . . . 6  |-  ( F 
C_  G  ->  ( A. s  e.  G  x  e.  ( ( cls `  J ) `  s )  ->  A. s  e.  F  x  e.  ( ( cls `  J
) `  s )
) )
31, 2syl 16 . . . . 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 999 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  J  e.  (TopOn `  X ) )
5 fclstopon 20358 . . . . . . . 8  |-  ( x  e.  ( J  fClus  G )  ->  ( J  e.  (TopOn `  X )  <->  G  e.  ( Fil `  X
) ) )
65adantl 466 . . . . . . 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 210 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  G  e.  ( Fil `  X ) )
8 isfcls2 20359 . . . . . 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 661 . . . . 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 1000 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  F  e.  ( Fil `  X ) )
11 isfcls2 20359 . . . . . 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 661 . . . . 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 268 . . . 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 434 . . 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 48 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  ( J  fClus  G )  ->  x  e.  ( J  fClus  F )
) )
1615ssrdv 3515 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( J  fClus  G )  C_  ( J  fClus  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767   A.wral 2817    C_ wss 3481   ` cfv 5593  (class class class)co 6294  TopOnctopon 19241   clsccl 19364   Filcfil 20191    fClus cfcls 20282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-int 4288  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-fbas 18263  df-topon 19248  df-fil 20192  df-fcls 20287
This theorem is referenced by:  fclsfnflim  20373  ufilcmp  20378  cnpfcfi  20386
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