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Theorem flfval 20357
Description: Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
flfval  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )

Proof of Theorem flfval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 toponmax 19296 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2 filtop 20222 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
3 elmapg 7431 . . . . 5  |-  ( ( X  e.  J  /\  Y  e.  L )  ->  ( F  e.  ( X  ^m  Y )  <-> 
F : Y --> X ) )
41, 2, 3syl2an 477 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( F  e.  ( X  ^m  Y )  <->  F : Y
--> X ) )
54biimpar 485 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  F : Y --> X )  ->  F  e.  ( X  ^m  Y ) )
6 flffval 20356 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fLimf  L )  =  ( f  e.  ( X  ^m  Y ) 
|->  ( J  fLim  (
( X  FilMap  f ) `
 L ) ) ) )
76fveq1d 5854 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  (
( J  fLimf  L ) `
 F )  =  ( ( f  e.  ( X  ^m  Y
)  |->  ( J  fLim  ( ( X  FilMap  f ) `
 L ) ) ) `  F ) )
8 oveq2 6285 . . . . . . 7  |-  ( f  =  F  ->  ( X  FilMap  f )  =  ( X  FilMap  F ) )
98fveq1d 5854 . . . . . 6  |-  ( f  =  F  ->  (
( X  FilMap  f ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
109oveq2d 6293 . . . . 5  |-  ( f  =  F  ->  ( J  fLim  ( ( X 
FilMap  f ) `  L
) )  =  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) )
11 eqid 2441 . . . . 5  |-  ( f  e.  ( X  ^m  Y )  |->  ( J 
fLim  ( ( X 
FilMap  f ) `  L
) ) )  =  ( f  e.  ( X  ^m  Y ) 
|->  ( J  fLim  (
( X  FilMap  f ) `
 L ) ) )
12 ovex 6305 . . . . 5  |-  ( J 
fLim  ( ( X 
FilMap  F ) `  L
) )  e.  _V
1310, 11, 12fvmpt 5937 . . . 4  |-  ( F  e.  ( X  ^m  Y )  ->  (
( f  e.  ( X  ^m  Y ) 
|->  ( J  fLim  (
( X  FilMap  f ) `
 L ) ) ) `  F )  =  ( J  fLim  ( ( X  FilMap  F ) `
 L ) ) )
147, 13sylan9eq 2502 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  F  e.  ( X  ^m  Y
) )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
155, 14syldan 470 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  F : Y --> X )  -> 
( ( J  fLimf  L ) `  F )  =  ( J  fLim  ( ( X  FilMap  F ) `
 L ) ) )
16153impa 1190 1  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    |-> cmpt 4491   -->wf 5570   ` cfv 5574  (class class class)co 6277    ^m cmap 7418  TopOnctopon 19262   Filcfil 20212    FilMap cfm 20300    fLim cflim 20301    fLimf cflf 20302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7420  df-fbas 18284  df-top 19266  df-topon 19269  df-fil 20213  df-flf 20307
This theorem is referenced by:  flfnei  20358  isflf  20360  hausflf  20364  flfcnp  20371  flfssfcf  20405  uffcfflf  20406  cnpfcf  20408  cnextcn  20433  tsmscls  20502  cnextucn  20672  cmetcaulem  21593  fmcncfil  27779
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