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Theorem cnpflf 18035
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 17316 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  (
( J  CnP  K
) `  A )
)  ->  F : X
--> Y )
213expa 1154 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
323adantl3 1116 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
4 cnpflfi 18033 . . . . . . 7  |-  ( ( A  e.  ( J 
fLim  f )  /\  F  e.  ( ( J  CnP  K ) `  A ) )  -> 
( F `  A
)  e.  ( ( K  fLimf  f ) `  F ) )
54expcom 426 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  A )  ->  ( A  e.  ( J  fLim  f )  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `  F ) ) )
65ralrimivw 2792 . . . . 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 454 . . . 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 520 . . 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 425 . 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 961 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  J  e.  (TopOn `  X )
)
11 simpl3 963 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  A  e.  X )
12 neiflim 18008 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
1310, 11, 12syl2anc 644 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
1411snssd 3945 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  { A }  C_  X )
15 snnzg 3923 . . . . . . . 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 17914 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  { A }  C_  X  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
1810, 14, 16, 17syl3anc 1185 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
19 oveq2 6091 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( J  fLim  f )  =  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
2019eleq2d 2505 . . . . . . . 8  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( A  e.  ( J  fLim  f )  <->  A  e.  ( J  fLim  ( ( nei `  J ) `
 { A }
) ) ) )
21 oveq2 6091 . . . . . . . . . 10  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( K  fLimf  f )  =  ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) )
2221fveq1d 5732 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( K  fLimf  f ) `
 F )  =  ( ( K  fLimf  ( ( nei `  J
) `  { A } ) ) `  F ) )
2322eleq2d 2505 . . . . . . . 8  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( F `  A
)  e.  ( ( K  fLimf  f ) `  F )  <->  ( F `  A )  e.  ( ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) `  F
) ) )
2420, 23imbi12d 313 . . . . . . 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 3050 . . . . . 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 40 . . . 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 676 . . 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 2438 . . . 4  |-  ( ( nei `  J ) `
 { A }
)  =  ( ( nei `  J ) `
 { A }
)
3029cnpflf2 18034 . . 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 227 . 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 185 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    C_ wss 3322   (/)c0 3630   {csn 3816   -->wf 5452   ` cfv 5456  (class class class)co 6083  TopOnctopon 16961   neicnei 17163    CnP ccnp 17291   Filcfil 17879    fLim cflim 17968    fLimf cflf 17969
This theorem is referenced by:  cnflf  18036  cnpfcf  18075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-map 7022  df-fbas 16701  df-fg 16702  df-top 16965  df-topon 16968  df-ntr 17086  df-nei 17164  df-cnp 17294  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974
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