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Theorem flfval 17781
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 16766 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2 filtop 17646 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
3 elmapg 6870 . . . . 5  |-  ( ( X  e.  J  /\  Y  e.  L )  ->  ( F  e.  ( X  ^m  Y )  <-> 
F : Y --> X ) )
41, 2, 3syl2an 463 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( F  e.  ( X  ^m  Y )  <->  F : Y
--> X ) )
54biimpar 471 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  F : Y --> X )  ->  F  e.  ( X  ^m  Y ) )
6 flffval 17780 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fLimf  L )  =  ( f  e.  ( X  ^m  Y ) 
|->  ( J  fLim  (
( X  FilMap  f ) `
 L ) ) ) )
76fveq1d 5607 . . . 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 5950 . . . . . . 7  |-  ( f  =  F  ->  ( X  FilMap  f )  =  ( X  FilMap  F ) )
98fveq1d 5607 . . . . . 6  |-  ( f  =  F  ->  (
( X  FilMap  f ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
109oveq2d 5958 . . . . 5  |-  ( f  =  F  ->  ( J  fLim  ( ( X 
FilMap  f ) `  L
) )  =  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) )
11 eqid 2358 . . . . 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 5967 . . . . 5  |-  ( J 
fLim  ( ( X 
FilMap  F ) `  L
) )  e.  _V
1310, 11, 12fvmpt 5682 . . . 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 2410 . . 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 456 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  F : Y --> X )  -> 
( ( J  fLimf  L ) `  F )  =  ( J  fLim  ( ( X  FilMap  F ) `
 L ) ) )
16153impa 1146 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    e. cmpt 4156   -->wf 5330   ` cfv 5334  (class class class)co 5942    ^m cmap 6857  TopOnctopon 16732   Filcfil 17636    FilMap cfm 17724    fLim cflim 17725    fLimf cflf 17726
This theorem is referenced by:  flfnei  17782  isflf  17784  hausflf  17788  flfcnp  17795  flfssfcf  17829  uffcfflf  17830  cnpfcf  17832  tsmscls  17916  cmetcaulem  18812  fmcncfil  23473  cnextcn  23504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-map 6859  df-fbas 16473  df-top 16736  df-topon 16739  df-fil 17637  df-flf 17731
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