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Theorem limcresi 12591
Description: Any limit of  F is also a limit of the restriction of  F. (Contributed by Mario Carneiro, 28-Dec-2016.)
Assertion
Ref Expression
limcresi  |-  ( F lim
CC  B )  C_  ( ( F  |`  C ) lim CC  B )

Proof of Theorem limcresi
Dummy variables  d  e  u  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limcrcl 12583 . . . . . . 7  |-  ( x  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
21simp1d 976 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  F : dom  F --> CC )
31simp2d 977 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  dom  F 
C_  CC )
41simp3d 978 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  B  e.  CC )
52, 3, 4ellimc3ap 12586 . . . . 5  |-  ( x  e.  ( F lim CC  B )  ->  (
x  e.  ( F lim
CC  B )  <->  ( x  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. u  e.  dom  F ( ( u #  B  /\  ( abs `  (
u  -  B ) )  <  d )  ->  ( abs `  (
( F `  u
)  -  x ) )  <  e ) ) ) )
65ibi 175 . . . 4  |-  ( x  e.  ( F lim CC  B )  ->  (
x  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. u  e. 
dom  F ( ( u #  B  /\  ( abs `  ( u  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 u )  -  x ) )  < 
e ) ) )
7 inss1 3262 . . . . . . . . 9  |-  ( dom 
F  i^i  C )  C_ 
dom  F
8 ssralv 3127 . . . . . . . . 9  |-  ( ( dom  F  i^i  C
)  C_  dom  F  -> 
( A. u  e. 
dom  F ( ( u #  B  /\  ( abs `  ( u  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 u )  -  x ) )  < 
e )  ->  A. u  e.  ( dom  F  i^i  C ) ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( F `  u )  -  x
) )  <  e
) ) )
97, 8ax-mp 7 . . . . . . . 8  |-  ( A. u  e.  dom  F ( ( u #  B  /\  ( abs `  ( u  -  B ) )  <  d )  -> 
( abs `  (
( F `  u
)  -  x ) )  <  e )  ->  A. u  e.  ( dom  F  i^i  C
) ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( F `  u )  -  x
) )  <  e
) )
10 elinel2 3229 . . . . . . . . . . . . . . 15  |-  ( u  e.  ( dom  F  i^i  C )  ->  u  e.  C )
11 fvres 5399 . . . . . . . . . . . . . . 15  |-  ( u  e.  C  ->  (
( F  |`  C ) `
 u )  =  ( F `  u
) )
1210, 11syl 14 . . . . . . . . . . . . . 14  |-  ( u  e.  ( dom  F  i^i  C )  ->  (
( F  |`  C ) `
 u )  =  ( F `  u
) )
1312adantl 273 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( F lim
CC  B )  /\  u  e.  ( dom  F  i^i  C ) )  ->  ( ( F  |`  C ) `  u
)  =  ( F `
 u ) )
1413fvoveq1d 5750 . . . . . . . . . . . 12  |-  ( ( x  e.  ( F lim
CC  B )  /\  u  e.  ( dom  F  i^i  C ) )  ->  ( abs `  (
( ( F  |`  C ) `  u
)  -  x ) )  =  ( abs `  ( ( F `  u )  -  x
) ) )
1514breq1d 3905 . . . . . . . . . . 11  |-  ( ( x  e.  ( F lim
CC  B )  /\  u  e.  ( dom  F  i^i  C ) )  ->  ( ( abs `  ( ( ( F  |`  C ) `  u
)  -  x ) )  <  e  <->  ( abs `  ( ( F `  u )  -  x
) )  <  e
) )
1615imbi2d 229 . . . . . . . . . 10  |-  ( ( x  e.  ( F lim
CC  B )  /\  u  e.  ( dom  F  i^i  C ) )  ->  ( ( ( u #  B  /\  ( abs `  ( u  -  B ) )  < 
d )  ->  ( abs `  ( ( ( F  |`  C ) `  u )  -  x
) )  <  e
)  <->  ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( F `  u )  -  x
) )  <  e
) ) )
1716biimprd 157 . . . . . . . . 9  |-  ( ( x  e.  ( F lim
CC  B )  /\  u  e.  ( dom  F  i^i  C ) )  ->  ( ( ( u #  B  /\  ( abs `  ( u  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 u )  -  x ) )  < 
e )  ->  (
( u #  B  /\  ( abs `  ( u  -  B ) )  <  d )  -> 
( abs `  (
( ( F  |`  C ) `  u
)  -  x ) )  <  e ) ) )
1817ralimdva 2473 . . . . . . . 8  |-  ( x  e.  ( F lim CC  B )  ->  ( A. u  e.  ( dom  F  i^i  C ) ( ( u #  B  /\  ( abs `  (
u  -  B ) )  <  d )  ->  ( abs `  (
( F `  u
)  -  x ) )  <  e )  ->  A. u  e.  ( dom  F  i^i  C
) ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( ( F  |`  C ) `  u
)  -  x ) )  <  e ) ) )
199, 18syl5 32 . . . . . . 7  |-  ( x  e.  ( F lim CC  B )  ->  ( A. u  e.  dom  F ( ( u #  B  /\  ( abs `  (
u  -  B ) )  <  d )  ->  ( abs `  (
( F `  u
)  -  x ) )  <  e )  ->  A. u  e.  ( dom  F  i^i  C
) ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( ( F  |`  C ) `  u
)  -  x ) )  <  e ) ) )
2019reximdv 2507 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  ( E. d  e.  RR+  A. u  e.  dom  F ( ( u #  B  /\  ( abs `  ( u  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 u )  -  x ) )  < 
e )  ->  E. d  e.  RR+  A. u  e.  ( dom  F  i^i  C ) ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( ( F  |`  C ) `  u
)  -  x ) )  <  e ) ) )
2120ralimdv 2474 . . . . 5  |-  ( x  e.  ( F lim CC  B )  ->  ( A. e  e.  RR+  E. d  e.  RR+  A. u  e. 
dom  F ( ( u #  B  /\  ( abs `  ( u  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 u )  -  x ) )  < 
e )  ->  A. e  e.  RR+  E. d  e.  RR+  A. u  e.  ( dom  F  i^i  C
) ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( ( F  |`  C ) `  u
)  -  x ) )  <  e ) ) )
2221anim2d 333 . . . 4  |-  ( x  e.  ( F lim CC  B )  ->  (
( x  e.  CC  /\ 
A. e  e.  RR+  E. d  e.  RR+  A. u  e.  dom  F ( ( u #  B  /\  ( abs `  ( u  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 u )  -  x ) )  < 
e ) )  -> 
( x  e.  CC  /\ 
A. e  e.  RR+  E. d  e.  RR+  A. u  e.  ( dom  F  i^i  C ) ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( ( F  |`  C ) `  u
)  -  x ) )  <  e ) ) ) )
236, 22mpd 13 . . 3  |-  ( x  e.  ( F lim CC  B )  ->  (
x  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. u  e.  ( dom  F  i^i  C ) ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( ( F  |`  C ) `  u
)  -  x ) )  <  e ) ) )
24 fresin 5259 . . . . 5  |-  ( F : dom  F --> CC  ->  ( F  |`  C ) : ( dom  F  i^i  C ) --> CC )
252, 24syl 14 . . . 4  |-  ( x  e.  ( F lim CC  B )  ->  ( F  |`  C ) : ( dom  F  i^i  C ) --> CC )
267, 3sstrid 3074 . . . 4  |-  ( x  e.  ( F lim CC  B )  ->  ( dom  F  i^i  C ) 
C_  CC )
2725, 26, 4ellimc3ap 12586 . . 3  |-  ( x  e.  ( F lim CC  B )  ->  (
x  e.  ( ( F  |`  C ) lim CC  B )  <->  ( x  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. u  e.  ( dom  F  i^i  C
) ( ( u #  B  /\  ( abs `  ( u  -  B
) )  <  d
)  ->  ( abs `  ( ( ( F  |`  C ) `  u
)  -  x ) )  <  e ) ) ) )
2823, 27mpbird 166 . 2  |-  ( x  e.  ( F lim CC  B )  ->  x  e.  ( ( F  |`  C ) lim CC  B ) )
2928ssriv 3067 1  |-  ( F lim
CC  B )  C_  ( ( F  |`  C ) lim CC  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2390   E.wrex 2391    i^i cin 3036    C_ wss 3037   class class class wbr 3895   dom cdm 4499    |` cres 4501   -->wf 5077   ` cfv 5081  (class class class)co 5728   CCcc 7545    < clt 7724    - cmin 7856   # cap 8261   RR+crp 9343   abscabs 10661   lim CC climc 12579
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pm 6499  df-limced 12581
This theorem is referenced by:  dvidlemap  12615  dvcnp2cntop  12618
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