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Theorem ufilcmp 18021
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 17893 . . . . . 6  |-  ( f  e.  ( UFil `  U. J )  ->  f  e.  ( Fil `  U. J ) )
2 eqid 2408 . . . . . . 7  |-  U. J  =  U. J
32fclscmpi 18018 . . . . . 6  |-  ( ( J  e.  Comp  /\  f  e.  ( Fil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
41, 3sylan2 461 . . . . 5  |-  ( ( J  e.  Comp  /\  f  e.  ( UFil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
54ralrimiva 2753 . . . 4  |-  ( J  e.  Comp  ->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) )
6 toponuni 16951 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
76fveq2d 5695 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  ( UFil `  X )  =  (
UFil `  U. J ) )
87raleqdv 2874 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) ) )
98adantl 453 . . . 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 213 . . 3  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  ->  A. f  e.  ( UFil `  X
) ( J  fClus  f )  =/=  (/) ) )
11 ufli 17903 . . . . . . 7  |-  ( ( X  e. UFL  /\  g  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) g  C_  f
)
1211adantlr 696 . . . . . 6  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) g  C_  f
)
13 r19.29 2810 . . . . . . 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 736 . . . . . . . . . . . . 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 732 . . . . . . . . . . . . 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 734 . . . . . . . . . . . . 13  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  ( f  e.  (
UFil `  X )  /\  g  C_  f ) )  ->  g  C_  f )
17 fclsss2 18012 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  g  e.  ( Fil `  X
)  /\  g  C_  f )  ->  ( J  fClus  f )  C_  ( J  fClus  g ) )
1814, 15, 16, 17syl3anc 1184 . . . . . . . . . . . 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 3624 . . . . . . . . . . . . 13  |-  ( ( ( J  fClus  f ) 
C_  ( J  fClus  g )  /\  ( J 
fClus  f )  =/=  (/) )  -> 
( J  fClus  g )  =/=  (/) )
2019ex 424 . . . . . . . . . . . 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 599 . . . . . . . . . 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 74 . . . . . . . . 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 421 . . . . . . . 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 2794 . . . . . . 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 30 . . . . . 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 656 . . . . 5  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  ( A. f  e.  ( UFil `  X ) ( J  fClus  f )  =/=  (/)  ->  ( J  fClus  g )  =/=  (/) ) )
2827ralrimdva 2760 . . . 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 18019 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  <->  A. g  e.  ( Fil `  X ) ( J  fClus  g )  =/=  (/) ) )
3029adantl 453 . . . 4  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. g  e.  ( Fil `  X ) ( J  fClus  g )  =/=  (/) ) )
3128, 30sylibrd 226 . . 3  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  ->  J  e.  Comp ) )
3210, 31impbid 184 . 2  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. f  e.  (
UFil `  X )
( J  fClus  f )  =/=  (/) ) )
33 uffclsflim 18020 . . . 4  |-  ( f  e.  ( UFil `  X
)  ->  ( J  fClus  f )  =  ( J  fLim  f )
)
3433neeq1d 2584 . . 3  |-  ( f  e.  ( UFil `  X
)  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fLim  f
)  =/=  (/) ) )
3534ralbiia 2702 . 2  |-  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( UFil `  X
) ( J  fLim  f )  =/=  (/) )
3632, 35syl6bb 253 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 177    /\ wa 359    e. wcel 1721    =/= wne 2571   A.wral 2670   E.wrex 2671    C_ wss 3284   (/)c0 3592   U.cuni 3979   ` cfv 5417  (class class class)co 6044  TopOnctopon 16918   Compccmp 17407   Filcfil 17834   UFilcufil 17888  UFLcufl 17889    fLim cflim 17923    fClus cfcls 17925
This theorem is referenced by:  alexsub  18033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-fi 7378  df-fbas 16658  df-fg 16659  df-top 16922  df-topon 16925  df-cld 17042  df-ntr 17043  df-cls 17044  df-nei 17121  df-cmp 17408  df-fil 17835  df-ufil 17890  df-ufl 17891  df-flim 17928  df-fcls 17930
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