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Theorem fclscmp 21045
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 2451 . . . . 5  |-  U. J  =  U. J
21fclscmpi 21044 . . . 4  |-  ( ( J  e.  Comp  /\  f  e.  ( Fil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
32ralrimiva 2802 . . 3  |-  ( J  e.  Comp  ->  A. f  e.  ( Fil `  U. J ) ( J 
fClus  f )  =/=  (/) )
4 toponuni 19942 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54fveq2d 5869 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
65raleqdv 2993 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( Fil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( Fil `  U. J ) ( J 
fClus  f )  =/=  (/) ) )
73, 6syl5ibr 225 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  ->  A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/) ) )
8 elpwi 3960 . . . . . 6  |-  ( x  e.  ~P ( Clsd `  J )  ->  x  C_  ( Clsd `  J
) )
9 vn0 3739 . . . . . . . . . 10  |-  _V  =/=  (/)
10 simpr 463 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  x  =  (/) )
1110inteqd 4239 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  |^| (/) )
12 int0 4248 . . . . . . . . . . . 12  |-  |^| (/)  =  _V
1311, 12syl6eq 2501 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  _V )
1413neeq1d 2683 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  ( |^| x  =/=  (/)  <->  _V  =/=  (/) ) )
159, 14mpbiri 237 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =/=  (/) )
1615a1d 26 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  ( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) )
17 vex 3048 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
18 ssfii 7933 . . . . . . . . . . . . . . . 16  |-  ( x  e.  _V  ->  x  C_  ( fi `  x
) )
1917, 18ax-mp 5 . . . . . . . . . . . . . . 15  |-  x  C_  ( fi `  x )
20 simplrl 770 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ( Clsd `  J
) )
211cldss2 20045 . . . . . . . . . . . . . . . . . . 19  |-  ( Clsd `  J )  C_  ~P U. J
224ad2antrr 732 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  X  =  U. J )
2322pweqd 3956 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  ~P X  =  ~P U. J )
2421, 23syl5sseqr 3481 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( Clsd `  J )  C_ 
~P X )
2520, 24sstrd 3442 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ~P X )
26 simpr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  =/=  (/) )
27 simplrr 771 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  -.  (/)  e.  ( fi
`  x ) )
28 toponmax 19943 . . . . . . . . . . . . . . . . . . 19  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2928ad2antrr 732 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  X  e.  J )
30 fsubbas 20882 . . . . . . . . . . . . . . . . . 18  |-  ( X  e.  J  ->  (
( fi `  x
)  e.  ( fBas `  X )  <->  ( x  C_ 
~P X  /\  x  =/=  (/)  /\  -.  (/)  e.  ( fi `  x ) ) ) )
3129, 30syl 17 . . . . . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . . . . 16  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( fi `  x
)  e.  ( fBas `  X ) )
33 ssfg 20887 . . . . . . . . . . . . . . . 16  |-  ( ( fi `  x )  e.  ( fBas `  X
)  ->  ( fi `  x )  C_  ( X filGen ( fi `  x ) ) )
3432, 33syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( fi `  x
)  C_  ( X filGen ( fi `  x
) ) )
3519, 34syl5ss 3443 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ( X filGen ( fi `  x ) ) )
3635sselda 3432 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( X filGen ( fi
`  x ) ) )
37 fclssscls 21033 . . . . . . . . . . . . 13  |-  ( y  e.  ( X filGen ( fi `  x ) )  ->  ( J  fClus  ( X filGen ( fi
`  x ) ) )  C_  ( ( cls `  J ) `  y ) )
3836, 37syl 17 . . . . . . . . . . . 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 3432 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( Clsd `  J
) )
40 cldcls 20057 . . . . . . . . . . . . 13  |-  ( y  e.  ( Clsd `  J
)  ->  ( ( cls `  J ) `  y )  =  y )
4139, 40syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  (
( cls `  J
) `  y )  =  y )
4238, 41sseqtrd 3468 . . . . . . . . . . 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 2802 . . . . . . . . . 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 4250 . . . . . . . . . 10  |-  ( ( J  fClus  ( X filGen ( fi `  x
) ) )  C_  |^| x  <->  A. y  e.  x  ( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  y )
4543, 44sylibr 216 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  |^| x )
46 fgcl 20893 . . . . . . . . . 10  |-  ( ( fi `  x )  e.  ( fBas `  X
)  ->  ( X filGen ( fi `  x
) )  e.  ( Fil `  X ) )
47 oveq2 6298 . . . . . . . . . . . 12  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( J  fClus  f )  =  ( J  fClus  ( X filGen ( fi `  x
) ) ) )
4847neeq1d 2683 . . . . . . . . . . 11  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fClus  ( X
filGen ( fi `  x
) ) )  =/=  (/) ) )
4948rspcv 3146 . . . . . . . . . 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 18 . . . . . . . . 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 3767 . . . . . . . . 9  |-  ( ( ( J  fClus  ( X
filGen ( fi `  x
) ) )  C_  |^| x  /\  ( J 
fClus  ( X filGen ( fi
`  x ) ) )  =/=  (/) )  ->  |^| x  =/=  (/) )
5245, 50, 51syl6an 548 . . . . . . . 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 2711 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  ->  ( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) )
5453expr 620 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  x  C_  ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  x )  -> 
( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) ) )
558, 54sylan2 477 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  ~P ( Clsd `  J
) )  ->  ( -.  (/)  e.  ( fi
`  x )  -> 
( A. f  e.  ( Fil `  X
) ( J  fClus  f )  =/=  (/)  ->  |^| x  =/=  (/) ) ) )
5655com23 81 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  ~P ( Clsd `  J
) )  ->  ( A. f  e.  ( Fil `  X ) ( J  fClus  f )  =/=  (/)  ->  ( -.  (/) 
e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
5756ralrimdva 2806 . . 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 19941 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
59 cmpfi 20423 . . . 4  |-  ( J  e.  Top  ->  ( J  e.  Comp  <->  A. x  e.  ~P  ( Clsd `  J
) ( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
6058, 59syl 17 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  <->  A. x  e.  ~P  ( Clsd `  J )
( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
6157, 60sylibrd 238 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( Fil `  X ) ( J 
fClus  f )  =/=  (/)  ->  J  e.  Comp ) )
627, 61impbid 194 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   _Vcvv 3045    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   U.cuni 4198   |^|cint 4234   ` cfv 5582  (class class class)co 6290   ficfi 7924   fBascfbas 18958   filGencfg 18959   Topctop 19917  TopOnctopon 19918   Clsdccld 20031   clsccl 20033   Compccmp 20401   Filcfil 20860    fClus cfcls 20951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-fbas 18967  df-fg 18968  df-top 19921  df-topon 19923  df-cld 20034  df-cls 20036  df-cmp 20402  df-fil 20861  df-fcls 20956
This theorem is referenced by:  ufilcmp  21047
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