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Theorem fclsss2 21049
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 1014 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  F  C_  G
)
2 ssralv 3461 . . . . . 6  |-  ( F 
C_  G  ->  ( A. s  e.  G  x  e.  ( ( cls `  J ) `  s )  ->  A. s  e.  F  x  e.  ( ( cls `  J
) `  s )
) )
31, 2syl 17 . . . . 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 1012 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  J  e.  (TopOn `  X ) )
5 fclstopon 21038 . . . . . . . 8  |-  ( x  e.  ( J  fClus  G )  ->  ( J  e.  (TopOn `  X )  <->  G  e.  ( Fil `  X
) ) )
65adantl 472 . . . . . . 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 215 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  G  e.  ( Fil `  X ) )
8 isfcls2 21039 . . . . . 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 671 . . . . 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 1013 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  F  e.  ( Fil `  X ) )
11 isfcls2 21039 . . . . . 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 671 . . . . 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 276 . . . 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 440 . . 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 50 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  ( J  fClus  G )  ->  x  e.  ( J  fClus  F )
) )
1615ssrdv 3406 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 189    /\ wa 375    /\ w3a 986    e. wcel 1891   A.wral 2737    C_ wss 3372   ` cfv 5561  (class class class)co 6276  TopOnctopon 19929   clsccl 20044   Filcfil 20871    fClus cfcls 20962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-int 4205  df-iin 4251  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5525  df-fun 5563  df-fn 5564  df-fv 5569  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-fbas 18978  df-topon 19934  df-fil 20872  df-fcls 20967
This theorem is referenced by:  fclsfnflim  21053  ufilcmp  21058  cnpfcfi  21066
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