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Theorem cnpflf 20671
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 19921 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  (
( J  CnP  K
) `  A )
)  ->  F : X
--> Y )
213expa 1194 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
323adantl3 1152 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  F : X
--> Y )
4 cnpflfi 20669 . . . . . . 7  |-  ( ( A  e.  ( J 
fLim  f )  /\  F  e.  ( ( J  CnP  K ) `  A ) )  -> 
( F `  A
)  e.  ( ( K  fLimf  f ) `  F ) )
54expcom 433 . . . . . 6  |-  ( F  e.  ( ( J  CnP  K ) `  A )  ->  ( A  e.  ( J  fLim  f )  ->  ( F `  A )  e.  ( ( K  fLimf  f ) `  F ) ) )
65ralrimivw 2869 . . . . 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 464 . . . 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 530 . . 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 432 . 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 997 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  J  e.  (TopOn `  X )
)
11 simpl3 999 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  A  e.  X )
12 neiflim 20644 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
1310, 11, 12syl2anc 659 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
1411snssd 4161 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  { A }  C_  X )
15 snnzg 4133 . . . . . . . 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 20550 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  { A }  C_  X  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
1810, 14, 16, 17syl3anc 1226 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X
)  /\  F : X
--> Y )  ->  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
19 oveq2 6278 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( J  fLim  f )  =  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
2019eleq2d 2524 . . . . . . . 8  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( A  e.  ( J  fLim  f )  <->  A  e.  ( J  fLim  ( ( nei `  J ) `
 { A }
) ) ) )
21 oveq2 6278 . . . . . . . . . 10  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  ( K  fLimf  f )  =  ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) )
2221fveq1d 5850 . . . . . . . . 9  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( K  fLimf  f ) `
 F )  =  ( ( K  fLimf  ( ( nei `  J
) `  { A } ) ) `  F ) )
2322eleq2d 2524 . . . . . . . 8  |-  ( f  =  ( ( nei `  J ) `  { A } )  ->  (
( F `  A
)  e.  ( ( K  fLimf  f ) `  F )  <->  ( F `  A )  e.  ( ( K  fLimf  ( ( nei `  J ) `
 { A }
) ) `  F
) ) )
2420, 23imbi12d 318 . . . . . . 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 3203 . . . . . 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 691 . . 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 2454 . . . 4  |-  ( ( nei `  J ) `
 { A }
)  =  ( ( nei `  J ) `
 { A }
)
3029cnpflf2 20670 . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804    C_ wss 3461   (/)c0 3783   {csn 4016   -->wf 5566   ` cfv 5570  (class class class)co 6270  TopOnctopon 19565   neicnei 19768    CnP ccnp 19896   Filcfil 20515    fLim cflim 20604    fLimf cflf 20605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-fbas 18614  df-fg 18615  df-top 19569  df-topon 19572  df-ntr 19691  df-nei 19769  df-cnp 19899  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610
This theorem is referenced by:  cnflf  20672  cnpfcf  20711
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