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Theorem cnpflf 20329
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 19557 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  (
( J  CnP  K
) `  A )
)  ->  F : X
--> Y )
213expa 1196 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
323adantl3 1154 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
4 cnpflfi 20327 . . . . . . 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 2879 . . . . 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 999 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  J  e.  (TopOn `  X )
)
11 simpl3 1001 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  A  e.  X )
12 neiflim 20302 . . . . . 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 4172 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  { A }  C_  X )
15 snnzg 4144 . . . . . . . 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 20208 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  { A }  C_  X  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
1810, 14, 16, 17syl3anc 1228 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
19 oveq2 6293 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( J  fLim  f )  =  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
2019eleq2d 2537 . . . . . . . 8  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( A  e.  ( J  fLim  f )  <->  A  e.  ( J  fLim  ( ( nei `  J ) `
 { A }
) ) ) )
21 oveq2 6293 . . . . . . . . . 10  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( K  fLimf  f )  =  ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) )
2221fveq1d 5868 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( K  fLimf  f ) `
 F )  =  ( ( K  fLimf  ( ( nei `  J
) `  { A } ) ) `  F ) )
2322eleq2d 2537 . . . . . . . 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 3210 . . . . . 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 2467 . . . 4  |-  ( ( nei `  J ) `
 { A }
)  =  ( ( nei `  J ) `
 { A }
)
3029cnpflf2 20328 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    C_ wss 3476   (/)c0 3785   {csn 4027   -->wf 5584   ` cfv 5588  (class class class)co 6285  TopOnctopon 19202   neicnei 19404    CnP ccnp 19532   Filcfil 20173    fLim cflim 20262    fLimf cflf 20263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-map 7423  df-fbas 18227  df-fg 18228  df-top 19206  df-topon 19209  df-ntr 19327  df-nei 19405  df-cnp 19535  df-fil 20174  df-fm 20266  df-flim 20267  df-flf 20268
This theorem is referenced by:  cnflf  20330  cnpfcf  20369
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