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Theorem uffclsflim 18016
Description: The cluster points of an ultrafilter are its limit points. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
uffclsflim  |-  ( F  e.  ( UFil `  X
)  ->  ( J  fClus  F )  =  ( J  fLim  F )
)

Proof of Theorem uffclsflim
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ufilfil 17889 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 fclsfnflim 18012 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  ( J  fClus  F )  <->  E. f  e.  ( Fil `  X ) ( F  C_  f  /\  x  e.  ( J  fLim  f ) ) ) )
31, 2syl 16 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  ( J  fClus  F )  <->  E. f  e.  ( Fil `  X ) ( F  C_  f  /\  x  e.  ( J  fLim  f ) ) ) )
43biimpa 471 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  ( J  fClus  F ) )  ->  E. f  e.  ( Fil `  X
) ( F  C_  f  /\  x  e.  ( J  fLim  f )
) )
5 simprrr 742 . . . . . 6  |-  ( ( ( F  e.  (
UFil `  X )  /\  x  e.  ( J  fClus  F ) )  /\  ( f  e.  ( Fil `  X
)  /\  ( F  C_  f  /\  x  e.  ( J  fLim  f
) ) ) )  ->  x  e.  ( J  fLim  f )
)
6 simpll 731 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  x  e.  ( J  fClus  F ) )  /\  ( f  e.  ( Fil `  X
)  /\  ( F  C_  f  /\  x  e.  ( J  fLim  f
) ) ) )  ->  F  e.  (
UFil `  X )
)
7 simprl 733 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  x  e.  ( J  fClus  F ) )  /\  ( f  e.  ( Fil `  X
)  /\  ( F  C_  f  /\  x  e.  ( J  fLim  f
) ) ) )  ->  f  e.  ( Fil `  X ) )
8 simprrl 741 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  x  e.  ( J  fClus  F ) )  /\  ( f  e.  ( Fil `  X
)  /\  ( F  C_  f  /\  x  e.  ( J  fLim  f
) ) ) )  ->  F  C_  f
)
9 ufilmax 17892 . . . . . . . 8  |-  ( ( F  e.  ( UFil `  X )  /\  f  e.  ( Fil `  X
)  /\  F  C_  f
)  ->  F  =  f )
106, 7, 8, 9syl3anc 1184 . . . . . . 7  |-  ( ( ( F  e.  (
UFil `  X )  /\  x  e.  ( J  fClus  F ) )  /\  ( f  e.  ( Fil `  X
)  /\  ( F  C_  f  /\  x  e.  ( J  fLim  f
) ) ) )  ->  F  =  f )
1110oveq2d 6056 . . . . . 6  |-  ( ( ( F  e.  (
UFil `  X )  /\  x  e.  ( J  fClus  F ) )  /\  ( f  e.  ( Fil `  X
)  /\  ( F  C_  f  /\  x  e.  ( J  fLim  f
) ) ) )  ->  ( J  fLim  F )  =  ( J 
fLim  f ) )
125, 11eleqtrrd 2481 . . . . 5  |-  ( ( ( F  e.  (
UFil `  X )  /\  x  e.  ( J  fClus  F ) )  /\  ( f  e.  ( Fil `  X
)  /\  ( F  C_  f  /\  x  e.  ( J  fLim  f
) ) ) )  ->  x  e.  ( J  fLim  F )
)
134, 12rexlimddv 2794 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  ( J  fClus  F ) )  ->  x  e.  ( J  fLim  F ) )
1413ex 424 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  ( J  fClus  F )  ->  x  e.  ( J  fLim  F )
) )
1514ssrdv 3314 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( J  fClus  F )  C_  ( J  fLim  F ) )
16 flimfcls 18011 . . 3  |-  ( J 
fLim  F )  C_  ( J  fClus  F )
1716a1i 11 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( J  fLim  F )  C_  ( J  fClus  F ) )
1815, 17eqssd 3325 1  |-  ( F  e.  ( UFil `  X
)  ->  ( J  fClus  F )  =  ( J  fLim  F )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667    C_ wss 3280   ` cfv 5413  (class class class)co 6040   Filcfil 17830   UFilcufil 17884    fLim cflim 17919    fClus cfcls 17921
This theorem is referenced by:  ufilcmp  18017  uffcfflf  18024
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-fin 7072  df-fi 7374  df-fbas 16654  df-fg 16655  df-top 16918  df-topon 16921  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-fil 17831  df-ufil 17886  df-flim 17924  df-fcls 17926
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