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Theorem flfval 18027
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 16998 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2 filtop 17892 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
3 elmapg 7034 . . . . 5  |-  ( ( X  e.  J  /\  Y  e.  L )  ->  ( F  e.  ( X  ^m  Y )  <-> 
F : Y --> X ) )
41, 2, 3syl2an 465 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( F  e.  ( X  ^m  Y )  <->  F : Y
--> X ) )
54biimpar 473 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  F : Y --> X )  ->  F  e.  ( X  ^m  Y ) )
6 flffval 18026 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fLimf  L )  =  ( f  e.  ( X  ^m  Y ) 
|->  ( J  fLim  (
( X  FilMap  f ) `
 L ) ) ) )
76fveq1d 5733 . . . 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 6092 . . . . . . 7  |-  ( f  =  F  ->  ( X  FilMap  f )  =  ( X  FilMap  F ) )
98fveq1d 5733 . . . . . 6  |-  ( f  =  F  ->  (
( X  FilMap  f ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
109oveq2d 6100 . . . . 5  |-  ( f  =  F  ->  ( J  fLim  ( ( X 
FilMap  f ) `  L
) )  =  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) )
11 eqid 2438 . . . . 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 6109 . . . . 5  |-  ( J 
fLim  ( ( X 
FilMap  F ) `  L
) )  e.  _V
1310, 11, 12fvmpt 5809 . . . 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 2490 . . 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 458 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  F : Y --> X )  -> 
( ( J  fLimf  L ) `  F )  =  ( J  fLim  ( ( X  FilMap  F ) `
 L ) ) )
16153impa 1149 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    e. cmpt 4269   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^m cmap 7021  TopOnctopon 16964   Filcfil 17882    FilMap cfm 17970    fLim cflim 17971    fLimf cflf 17972
This theorem is referenced by:  flfnei  18028  isflf  18030  hausflf  18034  flfcnp  18041  flfssfcf  18075  uffcfflf  18076  cnpfcf  18078  cnextcn  18103  tsmscls  18172  cnextucn  18338  cmetcaulem  19246  fmcncfil  24322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-fbas 16704  df-top 16968  df-topon 16971  df-fil 17883  df-flf 17977
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