MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fclsopni Unicode version

Theorem fclsopni 17968
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 2387 . . . . . . . . 9  |-  U. J  =  U. J
21fclsfil 17963 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  F  e.  ( Fil `  U. J
) )
3 fclstopon 17965 . . . . . . . 8  |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  U. J )  <-> 
F  e.  ( Fil `  U. J ) ) )
42, 3mpbird 224 . . . . . . 7  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  (TopOn `  U. J ) )
5 fclsopn 17967 . . . . . . 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 643 . . . . . 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 233 . . . . 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 450 . . . 4  |-  ( A  e.  ( J  fClus  F )  ->  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) )
9 eleq2 2448 . . . . . 6  |-  ( o  =  U  ->  ( A  e.  o  <->  A  e.  U ) )
10 ineq1 3478 . . . . . . . 8  |-  ( o  =  U  ->  (
o  i^i  s )  =  ( U  i^i  s ) )
1110neeq1d 2563 . . . . . . 7  |-  ( o  =  U  ->  (
( o  i^i  s
)  =/=  (/)  <->  ( U  i^i  s )  =/=  (/) ) )
1211ralbidv 2669 . . . . . 6  |-  ( o  =  U  ->  ( A. s  e.  F  ( o  i^i  s
)  =/=  (/)  <->  A. s  e.  F  ( U  i^i  s )  =/=  (/) ) )
139, 12imbi12d 312 . . . . 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 2992 . . . 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 3479 . . . . 5  |-  ( s  =  S  ->  ( U  i^i  s )  =  ( U  i^i  S
) )
1716neeq1d 2563 . . . 4  |-  ( s  =  S  ->  (
( U  i^i  s
)  =/=  (/)  <->  ( U  i^i  S )  =/=  (/) ) )
1817rspccv 2992 . . 3  |-  ( A. s  e.  F  ( U  i^i  s )  =/=  (/)  ->  ( S  e.  F  ->  ( U  i^i  S )  =/=  (/) ) )
1915, 18syl8 67 . 2  |-  ( A  e.  ( J  fClus  F )  ->  ( U  e.  J  ->  ( A  e.  U  ->  ( S  e.  F  ->  ( U  i^i  S )  =/=  (/) ) ) ) )
20193imp2 1168 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649    i^i cin 3262   (/)c0 3571   U.cuni 3957   ` cfv 5394  (class class class)co 6020  TopOnctopon 16882   Filcfil 17798    fClus cfcls 17889
This theorem is referenced by:  fclsneii  17970  supnfcls  17973  flimfnfcls  17981  cfilfcls  19098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-fbas 16623  df-top 16886  df-topon 16889  df-cld 17006  df-ntr 17007  df-cls 17008  df-fil 17799  df-fcls 17894
  Copyright terms: Public domain W3C validator