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Theorem fclsopni 19600
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 2443 . . . . . . . . 9  |-  U. J  =  U. J
21fclsfil 19595 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  F  e.  ( Fil `  U. J
) )
3 fclstopon 19597 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  U. J )  <-> 
F  e.  ( Fil `  U. J ) ) )
42, 3mpbird 232 . . . . . . 7  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  (TopOn `  U. J ) )
5 fclsopn 19599 . . . . . . 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 661 . . . . . 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 241 . . . . 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 463 . . . 4  |-  ( A  e.  ( J  fClus  F )  ->  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) )
9 eleq2 2504 . . . . . 6  |-  ( o  =  U  ->  ( A  e.  o  <->  A  e.  U ) )
10 ineq1 3557 . . . . . . . 8  |-  ( o  =  U  ->  (
o  i^i  s )  =  ( U  i^i  s ) )
1110neeq1d 2633 . . . . . . 7  |-  ( o  =  U  ->  (
( o  i^i  s
)  =/=  (/)  <->  ( U  i^i  s )  =/=  (/) ) )
1211ralbidv 2747 . . . . . 6  |-  ( o  =  U  ->  ( A. s  e.  F  ( o  i^i  s
)  =/=  (/)  <->  A. s  e.  F  ( U  i^i  s )  =/=  (/) ) )
139, 12imbi12d 320 . . . . 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 3082 . . . 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 16 . . 3  |-  ( A  e.  ( J  fClus  F )  ->  ( U  e.  J  ->  ( A  e.  U  ->  A. s  e.  F  ( U  i^i  s )  =/=  (/) ) ) )
16 ineq2 3558 . . . . 5  |-  ( s  =  S  ->  ( U  i^i  s )  =  ( U  i^i  S
) )
1716neeq1d 2633 . . . 4  |-  ( s  =  S  ->  (
( U  i^i  s
)  =/=  (/)  <->  ( U  i^i  S )  =/=  (/) ) )
1817rspccv 3082 . . 3  |-  ( A. s  e.  F  ( U  i^i  s )  =/=  (/)  ->  ( S  e.  F  ->  ( U  i^i  S )  =/=  (/) ) )
1915, 18syl8 70 . 2  |-  ( A  e.  ( J  fClus  F )  ->  ( U  e.  J  ->  ( A  e.  U  ->  ( S  e.  F  ->  ( U  i^i  S )  =/=  (/) ) ) ) )
20193imp2 1202 1  |-  ( ( A  e.  ( J 
fClus  F )  /\  ( U  e.  J  /\  A  e.  U  /\  S  e.  F )
)  ->  ( U  i^i  S )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727    i^i cin 3339   (/)c0 3649   U.cuni 4103   ` cfv 5430  (class class class)co 6103  TopOnctopon 18511   Filcfil 19430    fClus cfcls 19521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-fbas 17826  df-top 18515  df-topon 18518  df-cld 18635  df-ntr 18636  df-cls 18637  df-fil 19431  df-fcls 19526
This theorem is referenced by:  fclsneii  19602  supnfcls  19605  flimfnfcls  19613  cfilfcls  20797
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