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Theorem uffcfflf 20408
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 19298 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2 fmufil 20328 . . . 4  |-  ( ( X  e.  J  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( UFil `  X
) )
31, 2syl3an1 1261 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( X  FilMap  F ) `
 L )  e.  ( UFil `  X
) )
4 uffclsflim 20400 . . 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 20273 . . 3  |-  ( L  e.  ( UFil `  Y
)  ->  L  e.  ( Fil `  Y ) )
7 fcfval 20402 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( J  fClus  ( ( X  FilMap  F ) `  L ) ) )
86, 7syl3an2 1262 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( J  fClus  ( ( X  FilMap  F ) `  L ) ) )
9 flfval 20359 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
106, 9syl3an2 1262 . 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 2518 1  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( ( J  fLimf  L ) `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   -->wf 5590   ` cfv 5594  (class class class)co 6295  TopOnctopon 19264   Filcfil 20214   UFilcufil 20268    FilMap cfm 20302    fLim cflim 20303    fLimf cflf 20304    fClus cfcls 20305    fClusf cfcf 20306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-fin 7532  df-fi 7883  df-fbas 18286  df-fg 18287  df-top 19268  df-topon 19271  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-fil 20215  df-ufil 20270  df-fm 20307  df-flim 20308  df-flf 20309  df-fcls 20310  df-fcf 20311
This theorem is referenced by: (None)
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