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Theorem rlimres 13032
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 3567 . . . . . . . 8  |-  ( dom 
F  i^i  B )  C_ 
dom  F
2 ssralv 3413 . . . . . . . 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 2821 . . . . . 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 2789 . . . . 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 566 . . . 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 12975 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
9 rlimss 12976 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
10 eqidd 2442 . . . 4  |-  ( ( F  ~~> r  A  /\  z  e.  dom  F )  ->  ( F `  z )  =  ( F `  z ) )
118, 9, 10rlim 12969 . . 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 5575 . . . . . 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 657 . . . . 5  |-  ( F  ~~> r  A  ->  ( F  |`  ( dom  F  i^i  B ) ) : ( dom  F  i^i  B ) --> CC )
14 resres 5120 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
15 ffn 5556 . . . . . . . . 9  |-  ( F : dom  F --> CC  ->  F  Fn  dom  F )
16 fnresdm 5517 . . . . . . . . 9  |-  ( F  Fn  dom  F  -> 
( F  |`  dom  F
)  =  F )
178, 15, 163syl 20 . . . . . . . 8  |-  ( F  ~~> r  A  ->  ( F  |`  dom  F )  =  F )
1817reseq1d 5105 . . . . . . 7  |-  ( F  ~~> r  A  ->  (
( F  |`  dom  F
)  |`  B )  =  ( F  |`  B ) )
1914, 18syl5eqr 2487 . . . . . 6  |-  ( F  ~~> r  A  ->  ( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B ) )
2019feq1d 5543 . . . . 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 3364 . . . 4  |-  ( F  ~~> r  A  ->  ( dom  F  i^i  B ) 
C_  RR )
23 inss2 3568 . . . . . . 7  |-  ( dom 
F  i^i  B )  C_  B
2423sseli 3349 . . . . . 6  |-  ( z  e.  ( dom  F  i^i  B )  ->  z  e.  B )
25 fvres 5701 . . . . . 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 463 . . . 4  |-  ( ( F  ~~> r  A  /\  z  e.  ( dom  F  i^i  B ) )  ->  ( ( F  |`  B ) `  z
)  =  ( F `
 z ) )
2821, 22, 27rlim 12969 . . 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 1364    e. wcel 1761   A.wral 2713   E.wrex 2714    i^i cin 3324    C_ wss 3325   class class class wbr 4289   dom cdm 4836    |` cres 4838    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277    < clt 9414    <_ cle 9415    - cmin 9591   RR+crp 10987   abscabs 12719    ~~> r crli 12959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-pm 7213  df-rlim 12963
This theorem is referenced by:  rlimres2  13035  pnt  22806
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