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Theorem flfval 17983
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 16956 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2 filtop 17848 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
3 elmapg 6998 . . . . 5  |-  ( ( X  e.  J  /\  Y  e.  L )  ->  ( F  e.  ( X  ^m  Y )  <-> 
F : Y --> X ) )
41, 2, 3syl2an 464 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( F  e.  ( X  ^m  Y )  <->  F : Y
--> X ) )
54biimpar 472 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  F : Y --> X )  ->  F  e.  ( X  ^m  Y ) )
6 flffval 17982 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fLimf  L )  =  ( f  e.  ( X  ^m  Y ) 
|->  ( J  fLim  (
( X  FilMap  f ) `
 L ) ) ) )
76fveq1d 5697 . . . 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 6056 . . . . . . 7  |-  ( f  =  F  ->  ( X  FilMap  f )  =  ( X  FilMap  F ) )
98fveq1d 5697 . . . . . 6  |-  ( f  =  F  ->  (
( X  FilMap  f ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
109oveq2d 6064 . . . . 5  |-  ( f  =  F  ->  ( J  fLim  ( ( X 
FilMap  f ) `  L
) )  =  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) )
11 eqid 2412 . . . . 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 6073 . . . . 5  |-  ( J 
fLim  ( ( X 
FilMap  F ) `  L
) )  e.  _V
1310, 11, 12fvmpt 5773 . . . 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 2464 . . 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 457 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  F : Y --> X )  -> 
( ( J  fLimf  L ) `  F )  =  ( J  fLim  ( ( X  FilMap  F ) `
 L ) ) )
16153impa 1148 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    e. cmpt 4234   -->wf 5417   ` cfv 5421  (class class class)co 6048    ^m cmap 6985  TopOnctopon 16922   Filcfil 17838    FilMap cfm 17926    fLim cflim 17927    fLimf cflf 17928
This theorem is referenced by:  flfnei  17984  isflf  17986  hausflf  17990  flfcnp  17997  flfssfcf  18031  uffcfflf  18032  cnpfcf  18034  cnextcn  18059  tsmscls  18128  cnextucn  18294  cmetcaulem  19202  fmcncfil  24278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-fbas 16662  df-top 16926  df-topon 16929  df-fil 17839  df-flf 17933
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