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Theorem fclscmp 19745
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 2454 . . . . 5  |-  U. J  =  U. J
21fclscmpi 19744 . . . 4  |-  ( ( J  e.  Comp  /\  f  e.  ( Fil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
32ralrimiva 2830 . . 3  |-  ( J  e.  Comp  ->  A. f  e.  ( Fil `  U. J ) ( J 
fClus  f )  =/=  (/) )
4 toponuni 18674 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
54fveq2d 5806 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
65raleqdv 3029 . . 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 3980 . . . . . 6  |-  ( x  e.  ~P ( Clsd `  J )  ->  x  C_  ( Clsd `  J
) )
9 vn0 3755 . . . . . . . . . 10  |-  _V  =/=  (/)
10 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  x  =  (/) )
1110inteqd 4244 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  |^| (/) )
12 int0 4253 . . . . . . . . . . . 12  |-  |^| (/)  =  _V
1311, 12syl6eq 2511 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =  (/) )  ->  |^| x  =  _V )
1413neeq1d 2729 . . . . . . . . . 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 3081 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
18 ssfii 7784 . . . . . . . . . . . . . . . 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 18776 . . . . . . . . . . . . . . . . . . 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 3976 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  ~P X  =  ~P U. J )
2421, 23syl5sseqr 3516 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( Clsd `  J )  C_ 
~P X )
2520, 24sstrd 3477 . . . . . . . . . . . . . . . . 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 18675 . . . . . . . . . . . . . . . . . . 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 19582 . . . . . . . . . . . . . . . . . 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 1171 . . . . . . . . . . . . . . . 16  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  -> 
( fi `  x
)  e.  ( fBas `  X ) )
33 ssfg 19587 . . . . . . . . . . . . . . . 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 3478 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  (
x  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  x ) ) )  /\  x  =/=  (/) )  ->  x  C_  ( X filGen ( fi `  x ) ) )
3635sselda 3467 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( X filGen ( fi
`  x ) ) )
37 fclssscls 19733 . . . . . . . . . . . . 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 3467 . . . . . . . . . . . . 13  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( x  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  x ) ) )  /\  x  =/=  (/) )  /\  y  e.  x )  ->  y  e.  ( Clsd `  J
) )
40 cldcls 18788 . . . . . . . . . . . . 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 3503 . . . . . . . . . . 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 2830 . . . . . . . . . 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 4255 . . . . . . . . . 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 19593 . . . . . . . . . 10  |-  ( ( fi `  x )  e.  ( fBas `  X
)  ->  ( X filGen ( fi `  x
) )  e.  ( Fil `  X ) )
47 oveq2 6211 . . . . . . . . . . . 12  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( J  fClus  f )  =  ( J  fClus  ( X filGen ( fi `  x
) ) ) )
4847neeq1d 2729 . . . . . . . . . . 11  |-  ( f  =  ( X filGen ( fi `  x ) )  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fClus  ( X
filGen ( fi `  x
) ) )  =/=  (/) ) )
4948rspcv 3175 . . . . . . . . . 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 3781 . . . . . . . . 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 2770 . . . . . . 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 2912 . . 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 18673 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
59 cmpfi 19153 . . . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   _Vcvv 3078    C_ wss 3439   (/)c0 3748   ~Pcpw 3971   U.cuni 4202   |^|cint 4239   ` cfv 5529  (class class class)co 6203   ficfi 7775   fBascfbas 17939   filGencfg 17940   Topctop 18640  TopOnctopon 18641   Clsdccld 18762   clsccl 18764   Compccmp 19131   Filcfil 19560    fClus cfcls 19651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fi 7776  df-fbas 17949  df-fg 17950  df-top 18645  df-topon 18648  df-cld 18765  df-cls 18767  df-cmp 19132  df-fil 19561  df-fcls 19656
This theorem is referenced by:  ufilcmp  19747
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