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Theorem uffclsflim 18063
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 17936 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 fclsfnflim 18059 . . . . . . 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 17939 . . . . . . . 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 6097 . . . . . 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 2513 . . . . 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 2834 . . . 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 3354 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( J  fClus  F )  C_  ( J  fLim  F ) )
16 flimfcls 18058 . . 3  |-  ( J 
fLim  F )  C_  ( J  fClus  F )
1716a1i 11 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( J  fLim  F )  C_  ( J  fClus  F ) )
1815, 17eqssd 3365 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 1652    e. wcel 1725   E.wrex 2706    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Filcfil 17877   UFilcufil 17931    fLim cflim 17966    fClus cfcls 17968
This theorem is referenced by:  ufilcmp  18064  uffcfflf  18071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-fin 7113  df-fi 7416  df-fbas 16699  df-fg 16700  df-top 16963  df-topon 16966  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-fil 17878  df-ufil 17933  df-flim 17971  df-fcls 17973
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