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Theorem fclsss1 18059
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 963 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  C_  K
)
2 ssralv 3409 . . . . . . 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 550 . . . . . 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 962 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  F  e.  ( Fil `  X ) )
6 fclstopon 18049 . . . . . . . 8  |-  ( x  e.  ( K  fClus  F )  ->  ( K  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X
) ) )
76adantl 454 . . . . . . 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 225 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  K  e.  (TopOn `  X ) )
9 fclsopn 18051 . . . . . 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 644 . . . . 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 961 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  e.  (TopOn `  X ) )
12 fclsopn 18051 . . . . . 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 644 . . . . 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 261 . . . 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 425 . . 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 47 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  ->  ( x  e.  ( K  fClus  F )  ->  x  e.  ( J  fClus  F )
) )
1716ssrdv 3356 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 178    /\ wa 360    /\ w3a 937    e. wcel 1726    =/= wne 2601   A.wral 2707    i^i cin 3321    C_ wss 3322   (/)c0 3630   ` cfv 5457  (class class class)co 6084  TopOnctopon 16964   Filcfil 17882    fClus cfcls 17973
This theorem is referenced by:  fclscf  18062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-fbas 16704  df-top 16968  df-topon 16971  df-cld 17088  df-ntr 17089  df-cls 17090  df-fil 17883  df-fcls 17978
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