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Theorem rlimres 13392
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 3714 . . . . . . . 8  |-  ( dom 
F  i^i  B )  C_ 
dom  F
2 ssralv 3560 . . . . . . . 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 2925 . . . . . 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 2850 . . . . 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 13335 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
9 rlimss 13336 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
10 eqidd 2458 . . . 4  |-  ( ( F  ~~> r  A  /\  z  e.  dom  F )  ->  ( F `  z )  =  ( F `  z ) )
118, 9, 10rlim 13329 . . 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 5757 . . . . . 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 5296 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
15 ffn 5737 . . . . . . . . 9  |-  ( F : dom  F --> CC  ->  F  Fn  dom  F )
16 fnresdm 5696 . . . . . . . . 9  |-  ( F  Fn  dom  F  -> 
( F  |`  dom  F
)  =  F )
178, 15, 163syl 20 . . . . . . . 8  |-  ( F  ~~> r  A  ->  ( F  |`  dom  F )  =  F )
1817reseq1d 5282 . . . . . . 7  |-  ( F  ~~> r  A  ->  (
( F  |`  dom  F
)  |`  B )  =  ( F  |`  B ) )
1914, 18syl5eqr 2512 . . . . . 6  |-  ( F  ~~> r  A  ->  ( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B ) )
2019feq1d 5723 . . . . 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 3510 . . . 4  |-  ( F  ~~> r  A  ->  ( dom  F  i^i  B ) 
C_  RR )
23 inss2 3715 . . . . . . 7  |-  ( dom 
F  i^i  B )  C_  B
2423sseli 3495 . . . . . 6  |-  ( z  e.  ( dom  F  i^i  B )  ->  z  e.  B )
25 fvres 5886 . . . . . 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 13329 . . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    i^i cin 3470    C_ wss 3471   class class class wbr 4456   dom cdm 5008    |` cres 5010    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508    < clt 9645    <_ cle 9646    - cmin 9824   RR+crp 11245   abscabs 13078    ~~> r crli 13319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-pm 7441  df-rlim 13323
This theorem is referenced by:  rlimres2  13395  pnt  23924
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