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Theorem uffcfflf 18071
Description: If the domain filter is an ultrafilter, the cluster points of the function are the limit points. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
uffcfflf  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( ( J  fLimf  L ) `  F ) )

Proof of Theorem uffcfflf
StepHypRef Expression
1 toponmax 16993 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2 fmufil 17991 . . . 4  |-  ( ( X  e.  J  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( UFil `  X
) )
31, 2syl3an1 1217 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( X  FilMap  F ) `
 L )  e.  ( UFil `  X
) )
4 uffclsflim 18063 . . 3  |-  ( ( ( X  FilMap  F ) `
 L )  e.  ( UFil `  X
)  ->  ( J  fClus  ( ( X  FilMap  F ) `  L ) )  =  ( J 
fLim  ( ( X 
FilMap  F ) `  L
) ) )
53, 4syl 16 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  ( J  fClus  ( ( X 
FilMap  F ) `  L
) )  =  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) )
6 ufilfil 17936 . . 3  |-  ( L  e.  ( UFil `  Y
)  ->  L  e.  ( Fil `  Y ) )
7 fcfval 18065 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( J  fClus  ( ( X  FilMap  F ) `  L ) ) )
86, 7syl3an2 1218 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( J  fClus  ( ( X  FilMap  F ) `  L ) ) )
9 flfval 18022 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
106, 9syl3an2 1218 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
115, 8, 103eqtr4d 2478 1  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( ( J  fLimf  L ) `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   -->wf 5450   ` cfv 5454  (class class class)co 6081  TopOnctopon 16959   Filcfil 17877   UFilcufil 17931    FilMap cfm 17965    fLim cflim 17966    fLimf cflf 17967    fClus cfcls 17968    fClusf cfcf 17969
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-map 7020  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-fm 17970  df-flim 17971  df-flf 17972  df-fcls 17973  df-fcf 17974
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