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Theorem cnpflf 17712
Description: Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
Assertion
Ref Expression
cnpflf  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
( F : X --> Y  /\  A. f  e.  ( Fil `  X
) ( A  e.  ( J  fLim  f
)  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `
 F ) ) ) ) )
Distinct variable groups:    A, f    f, X    f, Y    f, F    f, J    f, K

Proof of Theorem cnpflf
StepHypRef Expression
1 cnpf2 16996 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  (
( J  CnP  K
) `  A )
)  ->  F : X
--> Y )
213expa 1151 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
323adantl3 1113 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
4 cnpflfi 17710 . . . . . . 7  |-  ( ( A  e.  ( J 
fLim  f )  /\  F  e.  ( ( J  CnP  K ) `  A ) )  -> 
( F `  A
)  e.  ( ( K  fLimf  f ) `  F ) )
54expcom 424 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  A )  ->  ( A  e.  ( J  fLim  f )  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `  F ) ) )
65ralrimivw 2640 . . . . 5  |-  ( F  e.  ( ( J  CnP  K ) `  A )  ->  A. f  e.  ( Fil `  X
) ( A  e.  ( J  fLim  f
)  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `
 F ) ) )
76adantl 452 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  A. f  e.  ( Fil `  X
) ( A  e.  ( J  fLim  f
)  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `
 F ) ) )
83, 7jca 518 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  ( F : X --> Y  /\  A. f  e.  ( Fil `  X ) ( A  e.  ( J  fLim  f )  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `
 F ) ) ) )
98ex 423 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  A )  ->  ( F : X
--> Y  /\  A. f  e.  ( Fil `  X
) ( A  e.  ( J  fLim  f
)  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `
 F ) ) ) ) )
10 simpl1 958 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  J  e.  (TopOn `  X )
)
11 simpl3 960 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  A  e.  X )
12 neiflim 17685 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
1310, 11, 12syl2anc 642 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
1411snssd 3776 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  { A }  C_  X )
15 snnzg 3756 . . . . . . . 8  |-  ( A  e.  X  ->  { A }  =/=  (/) )
1611, 15syl 15 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  { A }  =/=  (/) )
17 neifil 17591 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  { A }  C_  X  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
1810, 14, 16, 17syl3anc 1182 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
19 oveq2 5882 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( J  fLim  f )  =  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
2019eleq2d 2363 . . . . . . . 8  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( A  e.  ( J  fLim  f )  <->  A  e.  ( J  fLim  ( ( nei `  J ) `
 { A }
) ) ) )
21 oveq2 5882 . . . . . . . . . 10  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( K  fLimf  f )  =  ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) )
2221fveq1d 5543 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( K  fLimf  f ) `
 F )  =  ( ( K  fLimf  ( ( nei `  J
) `  { A } ) ) `  F ) )
2322eleq2d 2363 . . . . . . . 8  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( F `  A
)  e.  ( ( K  fLimf  f ) `  F )  <->  ( F `  A )  e.  ( ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) `  F
) ) )
2420, 23imbi12d 311 . . . . . . 7  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( A  e.  ( J  fLim  f )  ->  ( F `  A
)  e.  ( ( K  fLimf  f ) `  F ) )  <->  ( A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) )  -> 
( F `  A
)  e.  ( ( K  fLimf  ( ( nei `  J ) `  { A } ) ) `
 F ) ) ) )
2524rspcv 2893 . . . . . 6  |-  ( ( ( nei `  J
) `  { A } )  e.  ( Fil `  X )  ->  ( A. f  e.  ( Fil `  X
) ( A  e.  ( J  fLim  f
)  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `
 F ) )  ->  ( A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) )  -> 
( F `  A
)  e.  ( ( K  fLimf  ( ( nei `  J ) `  { A } ) ) `
 F ) ) ) )
2618, 25syl 15 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  ( A. f  e.  ( Fil `  X ) ( A  e.  ( J 
fLim  f )  -> 
( F `  A
)  e.  ( ( K  fLimf  f ) `  F ) )  -> 
( A  e.  ( J  fLim  ( ( nei `  J ) `  { A } ) )  ->  ( F `  A )  e.  ( ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) `  F
) ) ) )
2713, 26mpid 37 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  ( A. f  e.  ( Fil `  X ) ( A  e.  ( J 
fLim  f )  -> 
( F `  A
)  e.  ( ( K  fLimf  f ) `  F ) )  -> 
( F `  A
)  e.  ( ( K  fLimf  ( ( nei `  J ) `  { A } ) ) `
 F ) ) )
2827imdistanda 674 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  ->  ( ( F : X --> Y  /\  A. f  e.  ( Fil `  X ) ( A  e.  ( J  fLim  f )  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `
 F ) ) )  ->  ( F : X --> Y  /\  ( F `  A )  e.  ( ( K  fLimf  ( ( nei `  J
) `  { A } ) ) `  F ) ) ) )
29 eqid 2296 . . . 4  |-  ( ( nei `  J ) `
 { A }
)  =  ( ( nei `  J ) `
 { A }
)
3029cnpflf2 17711 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
( F : X --> Y  /\  ( F `  A )  e.  ( ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) `  F
) ) ) )
3128, 30sylibrd 225 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  ->  ( ( F : X --> Y  /\  A. f  e.  ( Fil `  X ) ( A  e.  ( J  fLim  f )  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `
 F ) ) )  ->  F  e.  ( ( J  CnP  K ) `  A ) ) )
329, 31impbid 183 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
( F : X --> Y  /\  A. f  e.  ( Fil `  X
) ( A  e.  ( J  fLim  f
)  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `
 F ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   {csn 3653   -->wf 5267   ` cfv 5271  (class class class)co 5874  TopOnctopon 16648   neicnei 16850    CnP ccnp 16971   Filcfil 17556    fLim cflim 17645    fLimf cflf 17646
This theorem is referenced by:  cnflf  17713  cnpfcf  17752
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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