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Theorem fclsss1 20396
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 1002 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  C_  K
)
2 ssralv 3549 . . . . . . 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 1001 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  F  e.  ( Fil `  X ) )
6 fclstopon 20386 . . . . . . . 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 20388 . . . . . 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 1000 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  e.  (TopOn `  X ) )
12 fclsopn 20388 . . . . . 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 3495 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 974    e. wcel 1804    =/= wne 2638   A.wral 2793    i^i cin 3460    C_ wss 3461   (/)c0 3770   ` cfv 5578  (class class class)co 6281  TopOnctopon 19268   Filcfil 20219    fClus cfcls 20310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-fbas 18290  df-top 19272  df-topon 19275  df-cld 19393  df-ntr 19394  df-cls 19395  df-fil 20220  df-fcls 20315
This theorem is referenced by:  fclscf  20399
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