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Theorem ufilcmp 19732
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 19604 . . . . . 6  |-  ( f  e.  ( UFil `  U. J )  ->  f  e.  ( Fil `  U. J ) )
2 eqid 2452 . . . . . . 7  |-  U. J  =  U. J
32fclscmpi 19729 . . . . . 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 2827 . . . 4  |-  ( J  e.  Comp  ->  A. f  e.  ( UFil `  U. J ) ( J 
fClus  f )  =/=  (/) )
6 toponuni 18659 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
76fveq2d 5798 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  ( UFil `  X )  =  (
UFil `  U. J ) )
87raleqdv 3023 . . . . 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 19614 . . . . . . 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 2957 . . . . . . 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 19723 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  g  e.  ( Fil `  X
)  /\  g  C_  f )  ->  ( J  fClus  f )  C_  ( J  fClus  g ) )
1814, 15, 16, 17syl3anc 1219 . . . . . . . . . . . 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 3773 . . . . . . . . . . . . 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 2941 . . . . . . 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 2906 . . . 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 19730 . . . . 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 19731 . . . 4  |-  ( f  e.  ( UFil `  X
)  ->  ( J  fClus  f )  =  ( J  fLim  f )
)
3433neeq1d 2726 . . 3  |-  ( f  e.  ( UFil `  X
)  ->  ( ( J  fClus  f )  =/=  (/) 
<->  ( J  fLim  f
)  =/=  (/) ) )
3534ralbiia 2835 . 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 1758    =/= wne 2645   A.wral 2796   E.wrex 2797    C_ wss 3431   (/)c0 3740   U.cuni 4194   ` cfv 5521  (class class class)co 6195  TopOnctopon 18626   Compccmp 19116   Filcfil 19545   UFilcufil 19599  UFLcufl 19600    fLim cflim 19634    fClus cfcls 19636
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fi 7767  df-fbas 17934  df-fg 17935  df-top 18630  df-topon 18633  df-cld 18750  df-ntr 18751  df-cls 18752  df-nei 18829  df-cmp 19117  df-fil 19546  df-ufil 19601  df-ufl 19602  df-flim 19639  df-fcls 19641
This theorem is referenced by:  alexsub  19744
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