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Theorem fclsopni 17710
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 2283 . . . . . . . . 9  |-  U. J  =  U. J
21fclsfil 17705 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  F  e.  ( Fil `  U. J
) )
3 fclstopon 17707 . . . . . . . 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 17709 . . . . . . 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 2344 . . . . . 6  |-  ( o  =  U  ->  ( A  e.  o  <->  A  e.  U ) )
10 ineq1 3363 . . . . . . . 8  |-  ( o  =  U  ->  (
o  i^i  s )  =  ( U  i^i  s ) )
1110neeq1d 2459 . . . . . . 7  |-  ( o  =  U  ->  (
( o  i^i  s
)  =/=  (/)  <->  ( U  i^i  s )  =/=  (/) ) )
1211ralbidv 2563 . . . . . 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 2881 . . . 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 3364 . . . . 5  |-  ( s  =  S  ->  ( U  i^i  s )  =  ( U  i^i  S
) )
1716neeq1d 2459 . . . 4  |-  ( s  =  S  ->  (
( U  i^i  s
)  =/=  (/)  <->  ( U  i^i  S )  =/=  (/) ) )
1817rspccv 2881 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    i^i cin 3151   (/)c0 3455   U.cuni 3827   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632   Filcfil 17540    fClus cfcls 17631
This theorem is referenced by:  fclsneii  17712  supnfcls  17715  flimfnfcls  17723  cfilfcls  18700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-fbas 17520  df-fil 17541  df-fcls 17636
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