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Theorem rlimres 13356
Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimres  |-  ( F  ~~> r  A  ->  ( F  |`  B )  ~~> r  A
)

Proof of Theorem rlimres
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3723 . . . . . . . 8  |-  ( dom 
F  i^i  B )  C_ 
dom  F
2 ssralv 3569 . . . . . . . 8  |-  ( ( dom  F  i^i  B
)  C_  dom  F  -> 
( A. z  e. 
dom  F ( y  <_  z  ->  ( abs `  ( ( F `
 z )  -  A ) )  < 
x )  ->  A. z  e.  ( dom  F  i^i  B ) ( y  <_ 
z  ->  ( abs `  ( ( F `  z )  -  A
) )  <  x
) ) )
31, 2ax-mp 5 . . . . . . 7  |-  ( A. z  e.  dom  F ( y  <_  z  ->  ( abs `  ( ( F `  z )  -  A ) )  <  x )  ->  A. z  e.  ( dom  F  i^i  B ) ( y  <_  z  ->  ( abs `  (
( F `  z
)  -  A ) )  <  x ) )
43reximi 2935 . . . . . 6  |-  ( E. y  e.  RR  A. z  e.  dom  F ( y  <_  z  ->  ( abs `  ( ( F `  z )  -  A ) )  <  x )  ->  E. y  e.  RR  A. z  e.  ( dom 
F  i^i  B )
( y  <_  z  ->  ( abs `  (
( F `  z
)  -  A ) )  <  x ) )
54ralimi 2860 . . . . 5  |-  ( A. x  e.  RR+  E. y  e.  RR  A. z  e. 
dom  F ( y  <_  z  ->  ( abs `  ( ( F `
 z )  -  A ) )  < 
x )  ->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  ( dom  F  i^i  B ) ( y  <_ 
z  ->  ( abs `  ( ( F `  z )  -  A
) )  <  x
) )
65anim2i 569 . . . 4  |-  ( ( A  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR  A. z  e. 
dom  F ( y  <_  z  ->  ( abs `  ( ( F `
 z )  -  A ) )  < 
x ) )  -> 
( A  e.  CC  /\ 
A. x  e.  RR+  E. y  e.  RR  A. z  e.  ( dom  F  i^i  B ) ( y  <_  z  ->  ( abs `  ( ( F `  z )  -  A ) )  <  x ) ) )
76a1i 11 . . 3  |-  ( F  ~~> r  A  ->  (
( A  e.  CC  /\ 
A. x  e.  RR+  E. y  e.  RR  A. z  e.  dom  F ( y  <_  z  ->  ( abs `  ( ( F `  z )  -  A ) )  <  x ) )  ->  ( A  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR  A. z  e.  ( dom  F  i^i  B ) ( y  <_ 
z  ->  ( abs `  ( ( F `  z )  -  A
) )  <  x
) ) ) )
8 rlimf 13299 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
9 rlimss 13300 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
10 eqidd 2468 . . . 4  |-  ( ( F  ~~> r  A  /\  z  e.  dom  F )  ->  ( F `  z )  =  ( F `  z ) )
118, 9, 10rlim 13293 . . 3  |-  ( F  ~~> r  A  ->  ( F 
~~> r  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR  A. z  e. 
dom  F ( y  <_  z  ->  ( abs `  ( ( F `
 z )  -  A ) )  < 
x ) ) ) )
12 fssres 5756 . . . . . 6  |-  ( ( F : dom  F --> CC  /\  ( dom  F  i^i  B )  C_  dom  F )  ->  ( F  |`  ( dom  F  i^i  B ) ) : ( dom  F  i^i  B
) --> CC )
138, 1, 12sylancl 662 . . . . 5  |-  ( F  ~~> r  A  ->  ( F  |`  ( dom  F  i^i  B ) ) : ( dom  F  i^i  B ) --> CC )
14 resres 5291 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
15 ffn 5736 . . . . . . . . 9  |-  ( F : dom  F --> CC  ->  F  Fn  dom  F )
16 fnresdm 5695 . . . . . . . . 9  |-  ( F  Fn  dom  F  -> 
( F  |`  dom  F
)  =  F )
178, 15, 163syl 20 . . . . . . . 8  |-  ( F  ~~> r  A  ->  ( F  |`  dom  F )  =  F )
1817reseq1d 5277 . . . . . . 7  |-  ( F  ~~> r  A  ->  (
( F  |`  dom  F
)  |`  B )  =  ( F  |`  B ) )
1914, 18syl5eqr 2522 . . . . . 6  |-  ( F  ~~> r  A  ->  ( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B ) )
2019feq1d 5722 . . . . 5  |-  ( F  ~~> r  A  ->  (
( F  |`  ( dom  F  i^i  B ) ) : ( dom 
F  i^i  B ) --> CC 
<->  ( F  |`  B ) : ( dom  F  i^i  B ) --> CC ) )
2113, 20mpbid 210 . . . 4  |-  ( F  ~~> r  A  ->  ( F  |`  B ) : ( dom  F  i^i  B ) --> CC )
221, 9syl5ss 3520 . . . 4  |-  ( F  ~~> r  A  ->  ( dom  F  i^i  B ) 
C_  RR )
23 inss2 3724 . . . . . . 7  |-  ( dom 
F  i^i  B )  C_  B
2423sseli 3505 . . . . . 6  |-  ( z  e.  ( dom  F  i^i  B )  ->  z  e.  B )
25 fvres 5885 . . . . . 6  |-  ( z  e.  B  ->  (
( F  |`  B ) `
 z )  =  ( F `  z
) )
2624, 25syl 16 . . . . 5  |-  ( z  e.  ( dom  F  i^i  B )  ->  (
( F  |`  B ) `
 z )  =  ( F `  z
) )
2726adantl 466 . . . 4  |-  ( ( F  ~~> r  A  /\  z  e.  ( dom  F  i^i  B ) )  ->  ( ( F  |`  B ) `  z
)  =  ( F `
 z ) )
2821, 22, 27rlim 13293 . . 3  |-  ( F  ~~> r  A  ->  (
( F  |`  B )  ~~> r  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR  A. z  e.  ( dom  F  i^i  B ) ( y  <_ 
z  ->  ( abs `  ( ( F `  z )  -  A
) )  <  x
) ) ) )
297, 11, 283imtr4d 268 . 2  |-  ( F  ~~> r  A  ->  ( F 
~~> r  A  ->  ( F  |`  B )  ~~> r  A
) )
3029pm2.43i 47 1  |-  ( F  ~~> r  A  ->  ( F  |`  B )  ~~> r  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    i^i cin 3480    C_ wss 3481   class class class wbr 4452   dom cdm 5004    |` cres 5006    Fn wfn 5588   -->wf 5589   ` cfv 5593  (class class class)co 6294   CCcc 9500   RRcr 9501    < clt 9638    <_ cle 9639    - cmin 9815   RR+crp 11230   abscabs 13042    ~~> r crli 13283
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-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559
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-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-pm 7433  df-rlim 13287
This theorem is referenced by:  rlimres2  13359  pnt  23642
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