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

Theorem fclscmp 17983
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 2387 . . . . 5  |-  U. J  =  U. J
21fclscmpi 17982 . . . 4  |-  ( ( J  e.  Comp  /\  f  e.  ( Fil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
32ralrimiva 2732 . . 3  |-  ( J  e.  Comp  ->  A. f  e.  ( Fil `  U. J ) ( J 
fClus  f )  =/=  (/) )
4 toponuni 16915 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54fveq2d 5672 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
65raleqdv 2853 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( Fil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( Fil `  U. J ) ( J 
fClus  f )  =/=  (/) ) )
73, 6syl5ibr 213 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  ->  A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/) ) )
8 elpwi 3750 . . . . . 6  |-  ( x  e.  ~P ( Clsd `  J )  ->  x  C_  ( Clsd `  J
) )
9 vn0 3578 . . . . . . . . . 10  |-  _V  =/=  (/)
10 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  x  =  (/) )
1110inteqd 3997 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  |^| (/) )
12 int0 4006 . . . . . . . . . . . 12  |-  |^| (/)  =  _V
1311, 12syl6eq 2435 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  _V )
1413neeq1d 2563 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  ( |^| x  =/=  (/)  <->  _V  =/=  (/) ) )
159, 14mpbiri 225 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =/=  (/) )
1615a1d 23 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  ( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) )
17 vex 2902 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
18 ssfii 7359 . . . . . . . . . . . . . . . 16  |-  ( x  e.  _V  ->  x  C_  ( fi `  x
) )
1917, 18ax-mp 8 . . . . . . . . . . . . . . 15  |-  x  C_  ( fi `  x )
20 simplrl 737 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ( Clsd `  J
) )
211cldss2 17017 . . . . . . . . . . . . . . . . . . 19  |-  ( Clsd `  J )  C_  ~P U. J
224ad2antrr 707 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  X  =  U. J )
2322pweqd 3747 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  ~P X  =  ~P U. J )
2421, 23syl5sseqr 3340 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( Clsd `  J )  C_ 
~P X )
2520, 24sstrd 3301 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ~P X )
26 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  =/=  (/) )
27 simplrr 738 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  -.  (/)  e.  ( fi
`  x ) )
28 toponmax 16916 . . . . . . . . . . . . . . . . . . 19  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2928ad2antrr 707 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  X  e.  J )
30 fsubbas 17820 . . . . . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . . . . 16  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( fi `  x
)  e.  ( fBas `  X ) )
33 ssfg 17825 . . . . . . . . . . . . . . . 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 3302 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ( X filGen ( fi `  x ) ) )
3635sselda 3291 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( X filGen ( fi
`  x ) ) )
37 fclssscls 17971 . . . . . . . . . . . . 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 3291 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( Clsd `  J
) )
40 cldcls 17029 . . . . . . . . . . . . 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 3327 . . . . . . . . . . 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 2732 . . . . . . . . . 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 4008 . . . . . . . . . 10  |-  ( ( J  fClus  ( X filGen ( fi `  x
) ) )  C_  |^| x  <->  A. y  e.  x  ( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  y )
4543, 44sylibr 204 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  |^| x )
46 fgcl 17831 . . . . . . . . . 10  |-  ( ( fi `  x )  e.  ( fBas `  X
)  ->  ( X filGen ( fi `  x
) )  e.  ( Fil `  X ) )
47 oveq2 6028 . . . . . . . . . . . 12  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( J  fClus  f )  =  ( J  fClus  ( X filGen ( fi `  x
) ) ) )
4847neeq1d 2563 . . . . . . . . . . 11  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fClus  ( X
filGen ( fi `  x
) ) )  =/=  (/) ) )
4948rspcv 2991 . . . . . . . . . 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 19 . . . . . . . . 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 3603 . . . . . . . . 9  |-  ( ( ( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  |^| x  /\  ( J 
fClus  ( X filGen ( fi
`  x ) ) )  =/=  (/) )  ->  |^| x  =/=  (/) )
5245, 50, 51ee12an 1369 . . . . . . . 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 2628 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  ->  ( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) )
5453expr 599 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  x )  -> 
( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) ) )
558, 54sylan2 461 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  ~P ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  x )  -> 
( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) ) )
5655com23 74 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  ~P ( Clsd `  J
) )  ->  ( A. f  e.  ( Fil `  X ) ( J  fClus  f )  =/=  (/)  ->  ( -.  (/) 
e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
5756ralrimdva 2739 . . 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 16914 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
59 cmpfi 17393 . . . 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 226 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( Fil `  X ) ( J 
fClus  f )  =/=  (/)  ->  J  e.  Comp ) )
627, 61impbid 184 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  <->  A. f  e.  ( Fil `  X ) ( J  fClus  f )  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   _Vcvv 2899    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   U.cuni 3957   |^|cint 3992   ` cfv 5394  (class class class)co 6020   ficfi 7350   fBascfbas 16615   filGencfg 16616   Topctop 16881  TopOnctopon 16882   Clsdccld 17003   clsccl 17005   Compccmp 17371   Filcfil 17798    fClus cfcls 17889
This theorem is referenced by:  ufilcmp  17985
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-3or 937  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-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  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-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  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-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-fbas 16623  df-fg 16624  df-top 16886  df-topon 16889  df-cld 17006  df-cls 17008  df-cmp 17372  df-fil 17799  df-fcls 17894
  Copyright terms: Public domain W3C validator