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Theorem cnpflf 19586
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 18866 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  (
( J  CnP  K
) `  A )
)  ->  F : X
--> Y )
213expa 1187 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
323adantl3 1146 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
4 cnpflfi 19584 . . . . . . 7  |-  ( ( A  e.  ( J 
fLim  f )  /\  F  e.  ( ( J  CnP  K ) `  A ) )  -> 
( F `  A
)  e.  ( ( K  fLimf  f ) `  F ) )
54expcom 435 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  A )  ->  ( A  e.  ( J  fLim  f )  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `  F ) ) )
65ralrimivw 2812 . . . . 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 466 . . . 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 532 . . 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 434 . 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 991 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  J  e.  (TopOn `  X )
)
11 simpl3 993 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  A  e.  X )
12 neiflim 19559 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
1310, 11, 12syl2anc 661 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
1411snssd 4030 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  { A }  C_  X )
15 snnzg 4004 . . . . . . . 8  |-  ( A  e.  X  ->  { A }  =/=  (/) )
1611, 15syl 16 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  { A }  =/=  (/) )
17 neifil 19465 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  { A }  C_  X  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
1810, 14, 16, 17syl3anc 1218 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
19 oveq2 6111 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( J  fLim  f )  =  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
2019eleq2d 2510 . . . . . . . 8  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( A  e.  ( J  fLim  f )  <->  A  e.  ( J  fLim  ( ( nei `  J ) `
 { A }
) ) ) )
21 oveq2 6111 . . . . . . . . . 10  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( K  fLimf  f )  =  ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) )
2221fveq1d 5705 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( K  fLimf  f ) `
 F )  =  ( ( K  fLimf  ( ( nei `  J
) `  { A } ) ) `  F ) )
2322eleq2d 2510 . . . . . . . 8  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( F `  A
)  e.  ( ( K  fLimf  f ) `  F )  <->  ( F `  A )  e.  ( ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) `  F
) ) )
2420, 23imbi12d 320 . . . . . . 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 3081 . . . . . 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 16 . . . . 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 41 . . . 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 693 . . 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 2443 . . . 4  |-  ( ( nei `  J ) `
 { A }
)  =  ( ( nei `  J ) `
 { A }
)
3029cnpflf2 19585 . . 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 234 . 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 191 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727    C_ wss 3340   (/)c0 3649   {csn 3889   -->wf 5426   ` cfv 5430  (class class class)co 6103  TopOnctopon 18511   neicnei 18713    CnP ccnp 18841   Filcfil 19430    fLim cflim 19519    fLimf cflf 19520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-map 7228  df-fbas 17826  df-fg 17827  df-top 18515  df-topon 18518  df-ntr 18636  df-nei 18714  df-cnp 18844  df-fil 19431  df-fm 19523  df-flim 19524  df-flf 19525
This theorem is referenced by:  cnflf  19587  cnpfcf  19626
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