MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uffcfflf Structured version   Unicode version

Theorem uffcfflf 19754
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 18675 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2 fmufil 19674 . . . 4  |-  ( ( X  e.  J  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( UFil `  X
) )
31, 2syl3an1 1252 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( X  FilMap  F ) `
 L )  e.  ( UFil `  X
) )
4 uffclsflim 19746 . . 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 19619 . . 3  |-  ( L  e.  ( UFil `  Y
)  ->  L  e.  ( Fil `  Y ) )
7 fcfval 19748 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( J  fClus  ( ( X  FilMap  F ) `  L ) ) )
86, 7syl3an2 1253 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( J  fClus  ( ( X  FilMap  F ) `  L ) ) )
9 flfval 19705 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
106, 9syl3an2 1253 . 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 2505 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 965    = wceq 1370    e. wcel 1758   -->wf 5525   ` cfv 5529  (class class class)co 6203  TopOnctopon 18641   Filcfil 19560   UFilcufil 19614    FilMap cfm 19648    fLim cflim 19649    fLimf cflf 19650    fClus cfcls 19651    fClusf cfcf 19652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-fin 7427  df-fi 7776  df-fbas 17949  df-fg 17950  df-top 18645  df-topon 18648  df-cld 18765  df-ntr 18766  df-cls 18767  df-nei 18844  df-fil 19561  df-ufil 19616  df-fm 19653  df-flim 19654  df-flf 19655  df-fcls 19656  df-fcf 19657
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator