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Theorem flfval 17685
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 16666 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
2 filtop 17550 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
3 elmapg 6785 . . . . 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 17684 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fLimf  L )  =  ( f  e.  ( X  ^m  Y ) 
|->  ( J  fLim  (
( X  FilMap  f ) `
 L ) ) ) )
76fveq1d 5527 . . . 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 5866 . . . . . . 7  |-  ( f  =  F  ->  ( X  FilMap  f )  =  ( X  FilMap  F ) )
98fveq1d 5527 . . . . . 6  |-  ( f  =  F  ->  (
( X  FilMap  f ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
109oveq2d 5874 . . . . 5  |-  ( f  =  F  ->  ( J  fLim  ( ( X 
FilMap  f ) `  L
) )  =  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) )
11 eqid 2283 . . . . 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 5883 . . . . 5  |-  ( J 
fLim  ( ( X 
FilMap  F ) `  L
) )  e.  _V
1310, 11, 12fvmpt 5602 . . . 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 2335 . . 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 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772  TopOnctopon 16632   Filcfil 17540    FilMap cfm 17628    fLim cflim 17629    fLimf cflf 17630
This theorem is referenced by:  flfnei  17686  isflf  17688  hausflf  17692  flfcnp  17699  flfssfcf  17733  uffcfflf  17734  cnpfcf  17736  tsmscls  17820  cmetcaulem  18714  conttnf2  25562  cnpflf4  25564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-fbas 17520  df-fil 17541  df-flf 17635
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