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Theorem uffclsflim 17726
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 17599 . . . . . . 7  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
2 fclsfnflim 17722 . . . . . . 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 15 . . . . . 6  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  ( J  fClus  F )  <->  E. f  e.  ( Fil `  X ) ( F  C_  f  /\  x  e.  ( J  fLim  f ) ) ) )
43biimpa 470 . . . . 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 741 . . . . . . . 8  |-  ( ( ( 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 730 . . . . . . . . . 10  |-  ( ( ( 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 732 . . . . . . . . . 10  |-  ( ( ( 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 740 . . . . . . . . . 10  |-  ( ( ( 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 17602 . . . . . . . . . 10  |-  ( ( F  e.  ( UFil `  X )  /\  f  e.  ( Fil `  X
)  /\  F  C_  f
)  ->  F  =  f )
106, 7, 8, 9syl3anc 1182 . . . . . . . . 9  |-  ( ( ( F  e.  (
UFil `  X )  /\  x  e.  ( J  fClus  F ) )  /\  ( f  e.  ( Fil `  X
)  /\  ( F  C_  f  /\  x  e.  ( J  fLim  f
) ) ) )  ->  F  =  f )
1110oveq2d 5874 . . . . . . . 8  |-  ( ( ( 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 2360 . . . . . . 7  |-  ( ( ( 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 )
)
1312expr 598 . . . . . 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
) ) )
1413rexlimdva 2667 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  ( J  fClus  F ) )  ->  ( E. f  e.  ( Fil `  X ) ( F 
C_  f  /\  x  e.  ( J  fLim  f
) )  ->  x  e.  ( J  fLim  F
) ) )
154, 14mpd 14 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  x  e.  ( J  fClus  F ) )  ->  x  e.  ( J  fLim  F ) )
1615ex 423 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( x  e.  ( J  fClus  F )  ->  x  e.  ( J  fLim  F )
) )
1716ssrdv 3185 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( J  fClus  F )  C_  ( J  fLim  F ) )
18 flimfcls 17721 . . 3  |-  ( J 
fLim  F )  C_  ( J  fClus  F )
1918a1i 10 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( J  fLim  F )  C_  ( J  fClus  F ) )
2017, 19eqssd 3196 1  |-  ( F  e.  ( UFil `  X
)  ->  ( J  fClus  F )  =  ( J  fLim  F )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Filcfil 17540   UFilcufil 17594    fLim cflim 17629    fClus cfcls 17631
This theorem is referenced by:  ufilcmp  17727  uffcfflf  17734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-top 16636  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-fbas 17520  df-fg 17521  df-fil 17541  df-ufil 17596  df-flim 17634  df-fcls 17636
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