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Theorem fclsss1 20255
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 1001 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  C_  K
)
2 ssralv 3564 . . . . . . 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 1000 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  F  e.  ( Fil `  X ) )
6 fclstopon 20245 . . . . . . . 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 20247 . . . . . 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 999 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  J  C_  K
)  /\  x  e.  ( K  fClus  F ) )  ->  J  e.  (TopOn `  X ) )
12 fclsopn 20247 . . . . . 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 3510 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 973    e. wcel 1767    =/= wne 2662   A.wral 2814    i^i cin 3475    C_ wss 3476   (/)c0 3785   ` cfv 5586  (class class class)co 6282  TopOnctopon 19159   Filcfil 20078    fClus cfcls 20169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fbas 18184  df-top 19163  df-topon 19166  df-cld 19283  df-ntr 19284  df-cls 19285  df-fil 20079  df-fcls 20174
This theorem is referenced by:  fclscf  20258
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