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

Theorem ufilcmp 20399
Description: A space is compact iff every ultrafilter converges. (Contributed by Jeff Hankins, 11-Dec-2009.) (Proof shortened by Mario Carneiro, 12-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
ufilcmp  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) ) )
Distinct variable groups:    f, J    f, X

Proof of Theorem ufilcmp
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 ufilfil 20271 . . . . . 6  |-  ( f  e.  ( UFil `  U. J )  ->  f  e.  ( Fil `  U. J ) )
2 eqid 2441 . . . . . . 7  |-  U. J  =  U. J
32fclscmpi 20396 . . . . . 6  |-  ( ( J  e.  Comp  /\  f  e.  ( Fil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
41, 3sylan2 474 . . . . 5  |-  ( ( J  e.  Comp  /\  f  e.  ( UFil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
54ralrimiva 2855 . . . 4  |-  ( J  e.  Comp  ->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) )
6 toponuni 19295 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
76fveq2d 5856 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  ( UFil `  X )  =  (
UFil `  U. J ) )
87raleqdv 3044 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) ) )
98adantl 466 . . . 4  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) ) )
105, 9syl5ibr 221 . . 3  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  ->  A. f  e.  ( UFil `  X
) ( J  fClus  f )  =/=  (/) ) )
11 ufli 20281 . . . . . . 7  |-  ( ( X  e. UFL  /\  g  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) g  C_  f
)
1211adantlr 714 . . . . . 6  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) g  C_  f
)
13 r19.29 2976 . . . . . . 7  |-  ( ( A. f  e.  (
UFil `  X )
( J  fClus  f )  =/=  (/)  /\  E. f  e.  ( UFil `  X
) g  C_  f
)  ->  E. f  e.  ( UFil `  X
) ( ( J 
fClus  f )  =/=  (/)  /\  g  C_  f ) )
14 simpllr 758 . . . . . . . . . . . . 13  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  ( f  e.  (
UFil `  X )  /\  g  C_  f ) )  ->  J  e.  (TopOn `  X ) )
15 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  ( f  e.  (
UFil `  X )  /\  g  C_  f ) )  ->  g  e.  ( Fil `  X ) )
16 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  ( f  e.  (
UFil `  X )  /\  g  C_  f ) )  ->  g  C_  f )
17 fclsss2 20390 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  g  e.  ( Fil `  X
)  /\  g  C_  f )  ->  ( J  fClus  f )  C_  ( J  fClus  g ) )
1814, 15, 16, 17syl3anc 1227 . . . . . . . . . . . 12  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  ( f  e.  (
UFil `  X )  /\  g  C_  f ) )  ->  ( J  fClus  f )  C_  ( J  fClus  g ) )
19 ssn0 3800 . . . . . . . . . . . . 13  |-  ( ( ( J  fClus  f ) 
C_  ( J  fClus  g )  /\  ( J 
fClus  f )  =/=  (/) )  -> 
( J  fClus  g )  =/=  (/) )
2019ex 434 . . . . . . . . . . . 12  |-  ( ( J  fClus  f )  C_  ( J  fClus  g )  ->  ( ( J 
fClus  f )  =/=  (/)  ->  ( J  fClus  g )  =/=  (/) ) )
2118, 20syl 16 . . . . . . . . . . 11  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  ( f  e.  (
UFil `  X )  /\  g  C_  f ) )  ->  ( ( J  fClus  f )  =/=  (/)  ->  ( J  fClus  g )  =/=  (/) ) )
2221expr 615 . . . . . . . . . 10  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  f  e.  ( UFil `  X ) )  -> 
( g  C_  f  ->  ( ( J  fClus  f )  =/=  (/)  ->  ( J  fClus  g )  =/=  (/) ) ) )
2322com23 78 . . . . . . . . 9  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  f  e.  ( UFil `  X ) )  -> 
( ( J  fClus  f )  =/=  (/)  ->  (
g  C_  f  ->  ( J  fClus  g )  =/=  (/) ) ) )
2423impd 431 . . . . . . . 8  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  f  e.  ( UFil `  X ) )  -> 
( ( ( J 
fClus  f )  =/=  (/)  /\  g  C_  f )  ->  ( J  fClus  g )  =/=  (/) ) )
2524rexlimdva 2933 . . . . . . 7  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  ( E. f  e.  ( UFil `  X ) ( ( J  fClus  f )  =/=  (/)  /\  g  C_  f )  ->  ( J  fClus  g )  =/=  (/) ) )
2613, 25syl5 32 . . . . . 6  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  (
( A. f  e.  ( UFil `  X
) ( J  fClus  f )  =/=  (/)  /\  E. f  e.  ( UFil `  X ) g  C_  f )  ->  ( J  fClus  g )  =/=  (/) ) )
2712, 26mpan2d 674 . . . . 5  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  ( A. f  e.  ( UFil `  X ) ( J  fClus  f )  =/=  (/)  ->  ( J  fClus  g )  =/=  (/) ) )
2827ralrimdva 2859 . . . 4  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  ->  A. g  e.  ( Fil `  X
) ( J  fClus  g )  =/=  (/) ) )
29 fclscmp 20397 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  <->  A. g  e.  ( Fil `  X ) ( J  fClus  g )  =/=  (/) ) )
3029adantl 466 . . . 4  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. g  e.  ( Fil `  X ) ( J  fClus  g )  =/=  (/) ) )
3128, 30sylibrd 234 . . 3  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  ->  J  e.  Comp ) )
3210, 31impbid 191 . 2  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. f  e.  (
UFil `  X )
( J  fClus  f )  =/=  (/) ) )
33 uffclsflim 20398 . . . 4  |-  ( f  e.  ( UFil `  X
)  ->  ( J  fClus  f )  =  ( J  fLim  f )
)
3433neeq1d 2718 . . 3  |-  ( f  e.  ( UFil `  X
)  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fLim  f
)  =/=  (/) ) )
3534ralbiia 2871 . 2  |-  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( UFil `  X
) ( J  fLim  f )  =/=  (/) )
3632, 35syl6bb 261 1  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1802    =/= wne 2636   A.wral 2791   E.wrex 2792    C_ wss 3458   (/)c0 3767   U.cuni 4230   ` cfv 5574  (class class class)co 6277  TopOnctopon 19262   Compccmp 19752   Filcfil 20212   UFilcufil 20266  UFLcufl 20267    fLim cflim 20301    fClus cfcls 20303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fi 7869  df-fbas 18284  df-fg 18285  df-top 19266  df-topon 19269  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-cmp 19753  df-fil 20213  df-ufil 20268  df-ufl 20269  df-flim 20306  df-fcls 20308
This theorem is referenced by:  alexsub  20411
  Copyright terms: Public domain W3C validator