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Theorem fclsss1 18011
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 962 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  C_  K
)
2 ssralv 3371 . . . . . . 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 549 . . . . . 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 961 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  F  e.  ( Fil `  X ) )
6 fclstopon 18001 . . . . . . . 8  |-  ( x  e.  ( K  fClus  F )  ->  ( K  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X
) ) )
76adantl 453 . . . . . . 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 224 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  K  e.  (TopOn `  X ) )
9 fclsopn 18003 . . . . . 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 643 . . . . 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 960 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  e.  (TopOn `  X ) )
12 fclsopn 18003 . . . . . 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 643 . . . . 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 260 . . . 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 424 . . 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 46 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( x  e.  ( K  fClus  F )  ->  x  e.  ( J  fClus  F )
) )
1716ssrdv 3318 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( K  fClus  F )  C_  ( J  fClus  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1721    =/= wne 2571   A.wral 2670    i^i cin 3283    C_ wss 3284   (/)c0 3592   ` cfv 5417  (class class class)co 6044  TopOnctopon 16918   Filcfil 17834    fClus cfcls 17925
This theorem is referenced by:  fclscf  18014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-fbas 16658  df-top 16922  df-topon 16925  df-cld 17042  df-ntr 17043  df-cls 17044  df-fil 17835  df-fcls 17930
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