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Theorem ufilcmp 20618
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 20490 . . . . . 6  |-  ( f  e.  ( UFil `  U. J )  ->  f  e.  ( Fil `  U. J ) )
2 eqid 2382 . . . . . . 7  |-  U. J  =  U. J
32fclscmpi 20615 . . . . . 6  |-  ( ( J  e.  Comp  /\  f  e.  ( Fil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
41, 3sylan2 472 . . . . 5  |-  ( ( J  e.  Comp  /\  f  e.  ( UFil `  U. J ) )  -> 
( J  fClus  f )  =/=  (/) )
54ralrimiva 2796 . . . 4  |-  ( J  e.  Comp  ->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) )
6 toponuni 19513 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
76fveq2d 5778 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  ( UFil `  X )  =  (
UFil `  U. J ) )
87raleqdv 2985 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( A. f  e.  ( UFil `  X ) ( J 
fClus  f )  =/=  (/)  <->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) ) )
98adantl 464 . . . 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 20500 . . . . . . 7  |-  ( ( X  e. UFL  /\  g  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) g  C_  f
)
1211adantlr 712 . . . . . 6  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  E. f  e.  ( UFil `  X
) g  C_  f
)
13 r19.29 2917 . . . . . . 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 753 . . . . . . . . . . . . 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 755 . . . . . . . . . . . . 13  |-  ( ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  /\  g  e.  ( Fil `  X ) )  /\  ( f  e.  (
UFil `  X )  /\  g  C_  f ) )  ->  g  C_  f )
17 fclsss2 20609 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  g  e.  ( Fil `  X
)  /\  g  C_  f )  ->  ( J  fClus  f )  C_  ( J  fClus  g ) )
1814, 15, 16, 17syl3anc 1226 . . . . . . . . . . . 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 3745 . . . . . . . . . . . . 13  |-  ( ( ( J  fClus  f ) 
C_  ( J  fClus  g )  /\  ( J 
fClus  f )  =/=  (/) )  -> 
( J  fClus  g )  =/=  (/) )
2019ex 432 . . . . . . . . . . . 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 613 . . . . . . . . . 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 429 . . . . . . . 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 2874 . . . . . . 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 672 . . . . 5  |-  ( ( ( X  e. UFL  /\  J  e.  (TopOn `  X
) )  /\  g  e.  ( Fil `  X
) )  ->  ( A. f  e.  ( UFil `  X ) ( J  fClus  f )  =/=  (/)  ->  ( J  fClus  g )  =/=  (/) ) )
2827ralrimdva 2800 . . . 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 20616 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Comp  <->  A. g  e.  ( Fil `  X ) ( J  fClus  g )  =/=  (/) ) )
3029adantl 464 . . . 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 20617 . . . 4  |-  ( f  e.  ( UFil `  X
)  ->  ( J  fClus  f )  =  ( J  fLim  f )
)
3433neeq1d 2659 . . 3  |-  ( f  e.  ( UFil `  X
)  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fLim  f
)  =/=  (/) ) )
3534ralbiia 2812 . 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 367    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733    C_ wss 3389   (/)c0 3711   U.cuni 4163   ` cfv 5496  (class class class)co 6196  TopOnctopon 19480   Compccmp 19972   Filcfil 20431   UFilcufil 20485  UFLcufl 20486    fLim cflim 20520    fClus cfcls 20522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fi 7786  df-fbas 18529  df-fg 18530  df-top 19484  df-topon 19487  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-cmp 19973  df-fil 20432  df-ufil 20487  df-ufl 20488  df-flim 20525  df-fcls 20527
This theorem is referenced by:  alexsub  20630
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