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Theorem ufilcmp 18095
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 17967 . . . . . 6  |-  ( f  e.  ( UFil `  U. J )  ->  f  e.  ( Fil `  U. J ) )
2 eqid 2442 . . . . . . 7  |-  U. J  =  U. J
32fclscmpi 18092 . . . . . 6  |-  ( ( J  e.  Comp  /\  f  e.  ( Fil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
41, 3sylan2 462 . . . . 5  |-  ( ( J  e.  Comp  /\  f  e.  ( UFil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
54ralrimiva 2795 . . . 4  |-  ( J  e.  Comp  ->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) )
6 toponuni 17023 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
76fveq2d 5761 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  ( UFil `  X )  =  (
UFil `  U. J ) )
87raleqdv 2916 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) ) )
98adantl 454 . . . 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 214 . . 3  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  ->  A. f  e.  ( UFil `  X
) ( J  fClus  f )  =/=  (/) ) )
11 ufli 17977 . . . . . . 7  |-  ( ( X  e. UFL  /\  g  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) g  C_  f
)
1211adantlr 697 . . . . . 6  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) g  C_  f
)
13 r19.29 2852 . . . . . . 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 737 . . . . . . . . . . . . 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 733 . . . . . . . . . . . . 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 735 . . . . . . . . . . . . 13  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  ( f  e.  (
UFil `  X )  /\  g  C_  f ) )  ->  g  C_  f )
17 fclsss2 18086 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  g  e.  ( Fil `  X
)  /\  g  C_  f )  ->  ( J  fClus  f )  C_  ( J  fClus  g ) )
1814, 15, 16, 17syl3anc 1185 . . . . . . . . . . . 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 3645 . . . . . . . . . . . . 13  |-  ( ( ( J  fClus  f ) 
C_  ( J  fClus  g )  /\  ( J 
fClus  f )  =/=  (/) )  -> 
( J  fClus  g )  =/=  (/) )
2019ex 425 . . . . . . . . . . . 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 600 . . . . . . . . . 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 75 . . . . . . . . 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 )  =/=  (/) ) ) )
2423imp3a 422 . . . . . . . 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 2836 . . . . . . 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 31 . . . . . 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 657 . . . . 5  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  ( A. f  e.  ( UFil `  X ) ( J  fClus  f )  =/=  (/)  ->  ( J  fClus  g )  =/=  (/) ) )
2827ralrimdva 2802 . . . 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 18093 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  <->  A. g  e.  ( Fil `  X ) ( J  fClus  g )  =/=  (/) ) )
3029adantl 454 . . . 4  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. g  e.  ( Fil `  X ) ( J  fClus  g )  =/=  (/) ) )
3128, 30sylibrd 227 . . 3  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  ->  J  e.  Comp ) )
3210, 31impbid 185 . 2  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. f  e.  (
UFil `  X )
( J  fClus  f )  =/=  (/) ) )
33 uffclsflim 18094 . . . 4  |-  ( f  e.  ( UFil `  X
)  ->  ( J  fClus  f )  =  ( J  fLim  f )
)
3433neeq1d 2620 . . 3  |-  ( f  e.  ( UFil `  X
)  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fLim  f
)  =/=  (/) ) )
3534ralbiia 2743 . 2  |-  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( UFil `  X
) ( J  fLim  f )  =/=  (/) )
3632, 35syl6bb 254 1  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1727    =/= wne 2605   A.wral 2711   E.wrex 2712    C_ wss 3306   (/)c0 3613   U.cuni 4039   ` cfv 5483  (class class class)co 6110  TopOnctopon 16990   Compccmp 17480   Filcfil 17908   UFilcufil 17962  UFLcufl 17963    fLim cflim 17997    fClus cfcls 17999
This theorem is referenced by:  alexsub  18107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-fi 7445  df-fbas 16730  df-fg 16731  df-top 16994  df-topon 16997  df-cld 17114  df-ntr 17115  df-cls 17116  df-nei 17193  df-cmp 17481  df-fil 17909  df-ufil 17964  df-ufl 17965  df-flim 18002  df-fcls 18004
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