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Theorem fclscmp 20399
Description: A space is compact iff every filter clusters. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclscmp  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  <->  A. f  e.  ( Fil `  X ) ( J  fClus  f )  =/=  (/) ) )
Distinct variable groups:    f, J    f, X

Proof of Theorem fclscmp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . 5  |-  U. J  =  U. J
21fclscmpi 20398 . . . 4  |-  ( ( J  e.  Comp  /\  f  e.  ( Fil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
32ralrimiva 2881 . . 3  |-  ( J  e.  Comp  ->  A. f  e.  ( Fil `  U. J ) ( J 
fClus  f )  =/=  (/) )
4 toponuni 19297 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54fveq2d 5876 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
65raleqdv 3069 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( Fil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( Fil `  U. J ) ( J 
fClus  f )  =/=  (/) ) )
73, 6syl5ibr 221 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  ->  A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/) ) )
8 elpwi 4025 . . . . . 6  |-  ( x  e.  ~P ( Clsd `  J )  ->  x  C_  ( Clsd `  J
) )
9 vn0 3797 . . . . . . . . . 10  |-  _V  =/=  (/)
10 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  x  =  (/) )
1110inteqd 4293 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  |^| (/) )
12 int0 4302 . . . . . . . . . . . 12  |-  |^| (/)  =  _V
1311, 12syl6eq 2524 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  _V )
1413neeq1d 2744 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  ( |^| x  =/=  (/)  <->  _V  =/=  (/) ) )
159, 14mpbiri 233 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =/=  (/) )
1615a1d 25 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  ( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) )
17 vex 3121 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
18 ssfii 7891 . . . . . . . . . . . . . . . 16  |-  ( x  e.  _V  ->  x  C_  ( fi `  x
) )
1917, 18ax-mp 5 . . . . . . . . . . . . . . 15  |-  x  C_  ( fi `  x )
20 simplrl 759 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ( Clsd `  J
) )
211cldss2 19399 . . . . . . . . . . . . . . . . . . 19  |-  ( Clsd `  J )  C_  ~P U. J
224ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  X  =  U. J )
2322pweqd 4021 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  ~P X  =  ~P U. J )
2421, 23syl5sseqr 3558 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( Clsd `  J )  C_ 
~P X )
2520, 24sstrd 3519 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ~P X )
26 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  =/=  (/) )
27 simplrr 760 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  -.  (/)  e.  ( fi
`  x ) )
28 toponmax 19298 . . . . . . . . . . . . . . . . . . 19  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2928ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  X  e.  J )
30 fsubbas 20236 . . . . . . . . . . . . . . . . . 18  |-  ( X  e.  J  ->  (
( fi `  x
)  e.  ( fBas `  X )  <->  ( x  C_ 
~P X  /\  x  =/=  (/)  /\  -.  (/)  e.  ( fi `  x ) ) ) )
3129, 30syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( ( fi `  x )  e.  (
fBas `  X )  <->  ( x  C_  ~P X  /\  x  =/=  (/)  /\  -.  (/) 
e.  ( fi `  x ) ) ) )
3225, 26, 27, 31mpbir3and 1179 . . . . . . . . . . . . . . . 16  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( fi `  x
)  e.  ( fBas `  X ) )
33 ssfg 20241 . . . . . . . . . . . . . . . 16  |-  ( ( fi `  x )  e.  ( fBas `  X
)  ->  ( fi `  x )  C_  ( X filGen ( fi `  x ) ) )
3432, 33syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( fi `  x
)  C_  ( X filGen ( fi `  x
) ) )
3519, 34syl5ss 3520 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ( X filGen ( fi `  x ) ) )
3635sselda 3509 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( X filGen ( fi
`  x ) ) )
37 fclssscls 20387 . . . . . . . . . . . . 13  |-  ( y  e.  ( X filGen ( fi `  x ) )  ->  ( J  fClus  ( X filGen ( fi
`  x ) ) )  C_  ( ( cls `  J ) `  y ) )
3836, 37syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  ( J  fClus  ( X filGen ( fi `  x ) ) )  C_  (
( cls `  J
) `  y )
)
3920sselda 3509 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( Clsd `  J
) )
40 cldcls 19411 . . . . . . . . . . . . 13  |-  ( y  e.  ( Clsd `  J
)  ->  ( ( cls `  J ) `  y )  =  y )
4139, 40syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  (
( cls `  J
) `  y )  =  y )
4238, 41sseqtrd 3545 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  ( J  fClus  ( X filGen ( fi `  x ) ) )  C_  y
)
4342ralrimiva 2881 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  A. y  e.  x  ( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  y )
44 ssint 4304 . . . . . . . . . 10  |-  ( ( J  fClus  ( X filGen ( fi `  x
) ) )  C_  |^| x  <->  A. y  e.  x  ( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  y )
4543, 44sylibr 212 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  |^| x )
46 fgcl 20247 . . . . . . . . . 10  |-  ( ( fi `  x )  e.  ( fBas `  X
)  ->  ( X filGen ( fi `  x
) )  e.  ( Fil `  X ) )
47 oveq2 6303 . . . . . . . . . . . 12  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( J  fClus  f )  =  ( J  fClus  ( X filGen ( fi `  x
) ) ) )
4847neeq1d 2744 . . . . . . . . . . 11  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fClus  ( X
filGen ( fi `  x
) ) )  =/=  (/) ) )
4948rspcv 3215 . . . . . . . . . 10  |-  ( ( X filGen ( fi `  x ) )  e.  ( Fil `  X
)  ->  ( A. f  e.  ( Fil `  X ) ( J 
fClus  f )  =/=  (/)  ->  ( J  fClus  ( X filGen ( fi `  x ) ) )  =/=  (/) ) )
5032, 46, 493syl 20 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  ( J  fClus  ( X filGen ( fi `  x ) ) )  =/=  (/) ) )
51 ssn0 3823 . . . . . . . . 9  |-  ( ( ( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  |^| x  /\  ( J 
fClus  ( X filGen ( fi
`  x ) ) )  =/=  (/) )  ->  |^| x  =/=  (/) )
5245, 50, 51syl6an 545 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) )
5316, 52pm2.61dane 2785 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  ->  ( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) )
5453expr 615 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  x )  -> 
( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) ) )
558, 54sylan2 474 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  ~P ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  x )  -> 
( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) ) )
5655com23 78 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  ~P ( Clsd `  J
) )  ->  ( A. f  e.  ( Fil `  X ) ( J  fClus  f )  =/=  (/)  ->  ( -.  (/) 
e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
5756ralrimdva 2885 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( Fil `  X ) ( J 
fClus  f )  =/=  (/)  ->  A. x  e.  ~P  ( Clsd `  J
) ( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
58 topontop 19296 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
59 cmpfi 19776 . . . 4  |-  ( J  e.  Top  ->  ( J  e.  Comp  <->  A. x  e.  ~P  ( Clsd `  J
) ( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
6058, 59syl 16 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  <->  A. x  e.  ~P  ( Clsd `  J )
( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
6157, 60sylibrd 234 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( Fil `  X ) ( J 
fClus  f )  =/=  (/)  ->  J  e.  Comp ) )
627, 61impbid 191 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  <->  A. f  e.  ( Fil `  X ) ( J  fClus  f )  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   U.cuni 4251   |^|cint 4288   ` cfv 5594  (class class class)co 6295   ficfi 7882   fBascfbas 18276   filGencfg 18277   Topctop 19263  TopOnctopon 19264   Clsdccld 19385   clsccl 19387   Compccmp 19754   Filcfil 20214    fClus cfcls 20305
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-fbas 18286  df-fg 18287  df-top 19268  df-topon 19271  df-cld 19388  df-cls 19390  df-cmp 19755  df-fil 20215  df-fcls 20310
This theorem is referenced by:  ufilcmp  20401
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