Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  conttnf2 Unicode version

Theorem conttnf2 25562
Description:  F is continous at point  A iff  ( F `  A ) is a limit of the image filter of the neighborhoods of  A. (Contributed by FL, 7-Aug-2011.) (Revised by Mario Carneiro, 6-Aug-2015.)
Hypotheses
Ref Expression
conttnf2.1  |-  L  =  ( ( nei `  J
) `  { A } )
conttnf2.2  |-  X  = 
U. K
conttnf2.3  |-  Y  = 
U. J
Assertion
Ref Expression
conttnf2  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
( F `  A
)  e.  ( K 
fLim  ( ( X 
FilMap  F ) `  L
) ) ) )

Proof of Theorem conttnf2
StepHypRef Expression
1 conttnf2.3 . . . . 5  |-  Y  = 
U. J
21toptopon 16671 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  Y ) )
3 conttnf2.2 . . . . 5  |-  X  = 
U. K
43toptopon 16671 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  X ) )
5 biid 227 . . . 4  |-  ( A  e.  Y  <->  A  e.  Y )
6 conttnf2.1 . . . . 5  |-  L  =  ( ( nei `  J
) `  { A } )
76cnpflf2 17695 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  K  e.  (TopOn `  X )  /\  A  e.  Y
)  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
( F : Y --> X  /\  ( F `  A )  e.  ( ( K  fLimf  L ) `
 F ) ) ) )
82, 4, 5, 7syl3anb 1225 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  A  e.  Y )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <->  ( F : Y --> X  /\  ( F `  A )  e.  ( ( K  fLimf  L ) `  F ) ) ) )
983adant3r 1179 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
( F : Y --> X  /\  ( F `  A )  e.  ( ( K  fLimf  L ) `
 F ) ) ) )
10 simp3r 984 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  F : Y
--> X )
1110biantrurd 494 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  ( ( F `  A )  e.  ( ( K  fLimf  L ) `  F )  <-> 
( F : Y --> X  /\  ( F `  A )  e.  ( ( K  fLimf  L ) `
 F ) ) ) )
12 simp2 956 . . . . 5  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  K  e.  Top )
1312, 4sylib 188 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  K  e.  (TopOn `  X ) )
14 simp1 955 . . . . . . 7  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  J  e.  Top )
1514, 2sylib 188 . . . . . 6  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  J  e.  (TopOn `  Y ) )
16 snssi 3759 . . . . . . . 8  |-  ( A  e.  Y  ->  { A }  C_  Y )
1716adantr 451 . . . . . . 7  |-  ( ( A  e.  Y  /\  F : Y --> X )  ->  { A }  C_  Y )
18173ad2ant3 978 . . . . . 6  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  { A }  C_  Y )
19 snnzg 3743 . . . . . . . 8  |-  ( A  e.  Y  ->  { A }  =/=  (/) )
2019adantr 451 . . . . . . 7  |-  ( ( A  e.  Y  /\  F : Y --> X )  ->  { A }  =/=  (/) )
21203ad2ant3 978 . . . . . 6  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  { A }  =/=  (/) )
22 neifil 17575 . . . . . 6  |-  ( ( J  e.  (TopOn `  Y )  /\  { A }  C_  Y  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  Y ) )
2315, 18, 21, 22syl3anc 1182 . . . . 5  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  ( ( nei `  J ) `  { A } )  e.  ( Fil `  Y
) )
246, 23syl5eqel 2367 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  L  e.  ( Fil `  Y ) )
25 flfval 17685 . . . 4  |-  ( ( K  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( K  fLimf  L ) `
 F )  =  ( K  fLim  (
( X  FilMap  F ) `
 L ) ) )
2613, 24, 10, 25syl3anc 1182 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  ( ( K  fLimf  L ) `  F )  =  ( K  fLim  ( ( X  FilMap  F ) `  L ) ) )
2726eleq2d 2350 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  ( ( F `  A )  e.  ( ( K  fLimf  L ) `  F )  <-> 
( F `  A
)  e.  ( K 
fLim  ( ( X 
FilMap  F ) `  L
) ) ) )
289, 11, 273bitr2d 272 1  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
( F `  A
)  e.  ( K 
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    =/= wne 2446    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632   neicnei 16834    CnP ccnp 16955   Filcfil 17540    FilMap cfm 17628    fLim cflim 17629    fLimf cflf 17630
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-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-ntr 16757  df-nei 16835  df-cnp 16958  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635
  Copyright terms: Public domain W3C validator