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Theorem conttnf2 25665
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 16687 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  Y ) )
3 conttnf2.2 . . . . 5  |-  X  = 
U. K
43toptopon 16687 . . . 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 17711 . . . 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 3775 . . . . . . . 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 3756 . . . . . . . 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 17591 . . . . . 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 2380 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  L  e.  ( Fil `  Y ) )
25 flfval 17701 . . . 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 2363 . 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 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   -->wf 5267   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648   neicnei 16850    CnP ccnp 16971   Filcfil 17556    FilMap cfm 17644    fLim cflim 17645    fLimf cflf 17646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-top 16652  df-topon 16655  df-ntr 16773  df-nei 16851  df-cnp 16974  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651
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