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Theorem fclsss1 19708
Description: A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclsss1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( K  fClus  F )  C_  ( J  fClus  F ) )

Proof of Theorem fclsss1
Dummy variables  o 
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
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  C_  K
)
2 ssralv 3511 . . . . . . 7  |-  ( J 
C_  K  ->  ( A. o  e.  K  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) )  ->  A. o  e.  J  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) ) ) )
32anim2d 565 . . . . . 6  |-  ( J 
C_  K  ->  (
( x  e.  X  /\  A. o  e.  K  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) ) )  ->  ( x  e.  X  /\  A. o  e.  J  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) ) )
41, 3syl 16 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  ( (
x  e.  X  /\  A. o  e.  K  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) ) )  ->  ( x  e.  X  /\  A. o  e.  J  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) ) )
5 simpl2 992 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  F  e.  ( Fil `  X ) )
6 fclstopon 19698 . . . . . . . 8  |-  ( x  e.  ( K  fClus  F )  ->  ( K  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X
) ) )
76adantl 466 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  ( K  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X
) ) )
85, 7mpbird 232 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  K  e.  (TopOn `  X ) )
9 fclsopn 19700 . . . . . 6  |-  ( ( K  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  ( K 
fClus  F )  <->  ( x  e.  X  /\  A. o  e.  K  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) ) )
108, 5, 9syl2anc 661 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  ( x  e.  ( K  fClus  F )  <-> 
( x  e.  X  /\  A. o  e.  K  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) ) ) ) )
11 simpl1 991 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  e.  (TopOn `  X ) )
12 fclsopn 19700 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  ( J 
fClus  F )  <->  ( x  e.  X  /\  A. o  e.  J  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) ) )
1311, 5, 12syl2anc 661 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  ( x  e.  ( J  fClus  F )  <-> 
( x  e.  X  /\  A. o  e.  J  ( x  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) ) ) ) )
144, 10, 133imtr4d 268 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  ( x  e.  ( K  fClus  F )  ->  x  e.  ( J  fClus  F )
) )
1514ex 434 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( x  e.  ( K  fClus  F )  ->  ( x  e.  ( K  fClus  F )  ->  x  e.  ( J  fClus  F )
) ) )
1615pm2.43d 48 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( x  e.  ( K  fClus  F )  ->  x  e.  ( J  fClus  F )
) )
1716ssrdv 3457 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( K  fClus  F )  C_  ( J  fClus  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2642   A.wral 2793    i^i cin 3422    C_ wss 3423   (/)c0 3732   ` cfv 5513  (class class class)co 6187  TopOnctopon 18612   Filcfil 19531    fClus cfcls 19622
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 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-iin 4269  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-fbas 17920  df-top 18616  df-topon 18619  df-cld 18736  df-ntr 18737  df-cls 18738  df-fil 19532  df-fcls 19627
This theorem is referenced by:  fclscf  19711
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