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Theorem fclsss2 19738
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 993 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  F  C_  G
)
2 ssralv 3527 . . . . . 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 991 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  J  e.  (TopOn `  X ) )
5 fclstopon 19727 . . . . . . . 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 19728 . . . . . 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 992 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  F  C_  G
)  /\  x  e.  ( J  fClus  G ) )  ->  F  e.  ( Fil `  X ) )
11 isfcls2 19728 . . . . . 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 3473 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 965    e. wcel 1758   A.wral 2799    C_ wss 3439   ` cfv 5529  (class class class)co 6203  TopOnctopon 18641   clsccl 18764   Filcfil 19560    fClus cfcls 19651
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-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-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-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-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-fbas 17949  df-topon 18648  df-fil 19561  df-fcls 19656
This theorem is referenced by:  fclsfnflim  19742  ufilcmp  19747  cnpfcfi  19755
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