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Theorem fclsopni 17726
Description: An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclsopni  |-  ( ( A  e.  ( J 
fClus  F )  /\  ( U  e.  J  /\  A  e.  U  /\  S  e.  F )
)  ->  ( U  i^i  S )  =/=  (/) )

Proof of Theorem fclsopni
Dummy variables  o 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . . . . . 9  |-  U. J  =  U. J
21fclsfil 17721 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  F  e.  ( Fil `  U. J
) )
3 fclstopon 17723 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  U. J )  <-> 
F  e.  ( Fil `  U. J ) ) )
42, 3mpbird 223 . . . . . . 7  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  (TopOn `  U. J ) )
5 fclsopn 17725 . . . . . . 7  |-  ( ( J  e.  (TopOn `  U. J )  /\  F  e.  ( Fil `  U. J ) )  -> 
( A  e.  ( J  fClus  F )  <->  ( A  e.  U. J  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) ) ) ) )
64, 2, 5syl2anc 642 . . . . . 6  |-  ( A  e.  ( J  fClus  F )  ->  ( A  e.  ( J  fClus  F )  <-> 
( A  e.  U. J  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) ) )
76ibi 232 . . . . 5  |-  ( A  e.  ( J  fClus  F )  ->  ( A  e.  U. J  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) )
87simprd 449 . . . 4  |-  ( A  e.  ( J  fClus  F )  ->  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) )
9 eleq2 2357 . . . . . 6  |-  ( o  =  U  ->  ( A  e.  o  <->  A  e.  U ) )
10 ineq1 3376 . . . . . . . 8  |-  ( o  =  U  ->  (
o  i^i  s )  =  ( U  i^i  s ) )
1110neeq1d 2472 . . . . . . 7  |-  ( o  =  U  ->  (
( o  i^i  s
)  =/=  (/)  <->  ( U  i^i  s )  =/=  (/) ) )
1211ralbidv 2576 . . . . . 6  |-  ( o  =  U  ->  ( A. s  e.  F  ( o  i^i  s
)  =/=  (/)  <->  A. s  e.  F  ( U  i^i  s )  =/=  (/) ) )
139, 12imbi12d 311 . . . . 5  |-  ( o  =  U  ->  (
( A  e.  o  ->  A. s  e.  F  ( o  i^i  s
)  =/=  (/) )  <->  ( A  e.  U  ->  A. s  e.  F  ( U  i^i  s )  =/=  (/) ) ) )
1413rspccv 2894 . . . 4  |-  ( A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) )  ->  ( U  e.  J  ->  ( A  e.  U  ->  A. s  e.  F  ( U  i^i  s
)  =/=  (/) ) ) )
158, 14syl 15 . . 3  |-  ( A  e.  ( J  fClus  F )  ->  ( U  e.  J  ->  ( A  e.  U  ->  A. s  e.  F  ( U  i^i  s )  =/=  (/) ) ) )
16 ineq2 3377 . . . . 5  |-  ( s  =  S  ->  ( U  i^i  s )  =  ( U  i^i  S
) )
1716neeq1d 2472 . . . 4  |-  ( s  =  S  ->  (
( U  i^i  s
)  =/=  (/)  <->  ( U  i^i  S )  =/=  (/) ) )
1817rspccv 2894 . . 3  |-  ( A. s  e.  F  ( U  i^i  s )  =/=  (/)  ->  ( S  e.  F  ->  ( U  i^i  S )  =/=  (/) ) )
1915, 18syl8 65 . 2  |-  ( A  e.  ( J  fClus  F )  ->  ( U  e.  J  ->  ( A  e.  U  ->  ( S  e.  F  ->  ( U  i^i  S )  =/=  (/) ) ) ) )
20193imp2 1166 1  |-  ( ( A  e.  ( J 
fClus  F )  /\  ( U  e.  J  /\  A  e.  U  /\  S  e.  F )
)  ->  ( U  i^i  S )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    i^i cin 3164   (/)c0 3468   U.cuni 3843   ` cfv 5271  (class class class)co 5874  TopOnctopon 16648   Filcfil 17556    fClus cfcls 17647
This theorem is referenced by:  fclsneii  17728  supnfcls  17731  flimfnfcls  17739  cfilfcls  18716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-top 16652  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-fbas 17536  df-fil 17557  df-fcls 17652
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