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Theorem limcresi 14612
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 14604 . . . . . . 7  |-  ( x  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
21simp1d 1011 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  F : dom  F --> CC )
31simp2d 1012 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  dom  F 
C_  CC )
41simp3d 1013 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  B  e.  CC )
52, 3, 4ellimc3ap 14607 . . . . 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 176 . . . 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 3370 . . . . . . . . 9  |-  ( dom 
F  i^i  C )  C_ 
dom  F
8 ssralv 3234 . . . . . . . . 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 5 . . . . . . . 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 3337 . . . . . . . . . . . . . . 15  |-  ( u  e.  ( dom  F  i^i  C )  ->  u  e.  C )
11 fvres 5558 . . . . . . . . . . . . . . 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 277 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( F lim
CC  B )  /\  u  e.  ( dom  F  i^i  C ) )  ->  ( ( F  |`  C ) `  u
)  =  ( F `
 u ) )
1413fvoveq1d 5919 . . . . . . . . . . . 12  |-  ( ( x  e.  ( F lim
CC  B )  /\  u  e.  ( dom  F  i^i  C ) )  ->  ( abs `  (
( ( F  |`  C ) `  u
)  -  x ) )  =  ( abs `  ( ( F `  u )  -  x
) ) )
1514breq1d 4028 . . . . . . . . . . 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 230 . . . . . . . . . 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 158 . . . . . . . . 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 2557 . . . . . . . 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 2591 . . . . . 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 2558 . . . . 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 337 . . . 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 5413 . . . . 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 3181 . . . 4  |-  ( x  e.  ( F lim CC  B )  ->  ( dom  F  i^i  C ) 
C_  CC )
2725, 26, 4ellimc3ap 14607 . . 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 167 . 2  |-  ( x  e.  ( F lim CC  B )  ->  x  e.  ( ( F  |`  C ) lim CC  B ) )
2928ssriv 3174 1  |-  ( F lim
CC  B )  C_  ( ( F  |`  C ) lim CC  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469    i^i cin 3143    C_ wss 3144   class class class wbr 4018   dom cdm 4644    |` cres 4646   -->wf 5231   ` cfv 5235  (class class class)co 5897   CCcc 7840    < clt 8023    - cmin 8159   # cap 8569   RR+crp 9685   abscabs 11041   lim CC climc 14600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pm 6678  df-limced 14602
This theorem is referenced by:  dvidlemap  14637  dvcnp2cntop  14640  dvcoapbr  14648
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