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Theorem fclscmp 19445
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 2433 . . . . 5  |-  U. J  =  U. J
21fclscmpi 19444 . . . 4  |-  ( ( J  e.  Comp  /\  f  e.  ( Fil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
32ralrimiva 2789 . . 3  |-  ( J  e.  Comp  ->  A. f  e.  ( Fil `  U. J ) ( J 
fClus  f )  =/=  (/) )
4 toponuni 18374 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54fveq2d 5683 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
65raleqdv 2913 . . 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 3857 . . . . . 6  |-  ( x  e.  ~P ( Clsd `  J )  ->  x  C_  ( Clsd `  J
) )
9 vn0 3632 . . . . . . . . . 10  |-  _V  =/=  (/)
10 simpr 458 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  x  =  (/) )
1110inteqd 4121 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  |^| (/) )
12 int0 4130 . . . . . . . . . . . 12  |-  |^| (/)  =  _V
1311, 12syl6eq 2481 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  _V )
1413neeq1d 2611 . . . . . . . . . 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 2965 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
18 ssfii 7657 . . . . . . . . . . . . . . . 16  |-  ( x  e.  _V  ->  x  C_  ( fi `  x
) )
1917, 18ax-mp 5 . . . . . . . . . . . . . . 15  |-  x  C_  ( fi `  x )
20 simplrl 752 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ( Clsd `  J
) )
211cldss2 18476 . . . . . . . . . . . . . . . . . . 19  |-  ( Clsd `  J )  C_  ~P U. J
224ad2antrr 718 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  X  =  U. J )
2322pweqd 3853 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  ~P X  =  ~P U. J )
2421, 23syl5sseqr 3393 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( Clsd `  J )  C_ 
~P X )
2520, 24sstrd 3354 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ~P X )
26 simpr 458 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  =/=  (/) )
27 simplrr 753 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  -.  (/)  e.  ( fi
`  x ) )
28 toponmax 18375 . . . . . . . . . . . . . . . . . . 19  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2928ad2antrr 718 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  X  e.  J )
30 fsubbas 19282 . . . . . . . . . . . . . . . . . 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 1164 . . . . . . . . . . . . . . . 16  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( fi `  x
)  e.  ( fBas `  X ) )
33 ssfg 19287 . . . . . . . . . . . . . . . 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 3355 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ( X filGen ( fi `  x ) ) )
3635sselda 3344 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( X filGen ( fi
`  x ) ) )
37 fclssscls 19433 . . . . . . . . . . . . 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 3344 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( Clsd `  J
) )
40 cldcls 18488 . . . . . . . . . . . . 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 3380 . . . . . . . . . . 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 2789 . . . . . . . . . 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 4132 . . . . . . . . . 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 19293 . . . . . . . . . 10  |-  ( ( fi `  x )  e.  ( fBas `  X
)  ->  ( X filGen ( fi `  x
) )  e.  ( Fil `  X ) )
47 oveq2 6088 . . . . . . . . . . . 12  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( J  fClus  f )  =  ( J  fClus  ( X filGen ( fi `  x
) ) ) )
4847neeq1d 2611 . . . . . . . . . . 11  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fClus  ( X
filGen ( fi `  x
) ) )  =/=  (/) ) )
4948rspcv 3058 . . . . . . . . . 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 3658 . . . . . . . . 9  |-  ( ( ( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  |^| x  /\  ( J 
fClus  ( X filGen ( fi
`  x ) ) )  =/=  (/) )  ->  |^| x  =/=  (/) )
5245, 50, 51syl6an 540 . . . . . . . 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 2679 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  ->  ( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) )
5453expr 610 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  x )  -> 
( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) ) )
558, 54sylan2 471 . . . . 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 2796 . . 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 18373 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
59 cmpfi 18853 . . . 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596   A.wral 2705   _Vcvv 2962    C_ wss 3316   (/)c0 3625   ~Pcpw 3848   U.cuni 4079   |^|cint 4116   ` cfv 5406  (class class class)co 6080   ficfi 7648   fBascfbas 17648   filGencfg 17649   Topctop 18340  TopOnctopon 18341   Clsdccld 18462   clsccl 18464   Compccmp 18831   Filcfil 19260    fClus cfcls 19351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fi 7649  df-fbas 17658  df-fg 17659  df-top 18345  df-topon 18348  df-cld 18465  df-cls 18467  df-cmp 18832  df-fil 19261  df-fcls 19356
This theorem is referenced by:  ufilcmp  19447
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