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Theorem rlimres 12352
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 3561 . . . . . . . 8  |-  ( dom 
F  i^i  B )  C_ 
dom  F
2 ssralv 3407 . . . . . . . 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 8 . . . . . . 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 2813 . . . . . 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 2781 . . . . 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 553 . . . 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 12295 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
9 rlimss 12296 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
10 eqidd 2437 . . . 4  |-  ( ( F  ~~> r  A  /\  z  e.  dom  F )  ->  ( F `  z )  =  ( F `  z ) )
118, 9, 10rlim 12289 . . 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 5610 . . . . . 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 644 . . . . 5  |-  ( F  ~~> r  A  ->  ( F  |`  ( dom  F  i^i  B ) ) : ( dom  F  i^i  B ) --> CC )
14 resres 5159 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
15 ffn 5591 . . . . . . . . 9  |-  ( F : dom  F --> CC  ->  F  Fn  dom  F )
16 fnresdm 5554 . . . . . . . . 9  |-  ( F  Fn  dom  F  -> 
( F  |`  dom  F
)  =  F )
178, 15, 163syl 19 . . . . . . . 8  |-  ( F  ~~> r  A  ->  ( F  |`  dom  F )  =  F )
1817reseq1d 5145 . . . . . . 7  |-  ( F  ~~> r  A  ->  (
( F  |`  dom  F
)  |`  B )  =  ( F  |`  B ) )
1914, 18syl5eqr 2482 . . . . . 6  |-  ( F  ~~> r  A  ->  ( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B ) )
2019feq1d 5580 . . . . 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 202 . . . 4  |-  ( F  ~~> r  A  ->  ( F  |`  B ) : ( dom  F  i^i  B ) --> CC )
221, 9syl5ss 3359 . . . 4  |-  ( F  ~~> r  A  ->  ( dom  F  i^i  B ) 
C_  RR )
23 inss2 3562 . . . . . . 7  |-  ( dom 
F  i^i  B )  C_  B
2423sseli 3344 . . . . . 6  |-  ( z  e.  ( dom  F  i^i  B )  ->  z  e.  B )
25 fvres 5745 . . . . . 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 453 . . . 4  |-  ( ( F  ~~> r  A  /\  z  e.  ( dom  F  i^i  B ) )  ->  ( ( F  |`  B ) `  z
)  =  ( F `
 z ) )
2821, 22, 27rlim 12289 . . 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 260 . 2  |-  ( F  ~~> r  A  ->  ( F 
~~> r  A  ->  ( F  |`  B )  ~~> r  A
) )
3029pm2.43i 45 1  |-  ( F  ~~> r  A  ->  ( F  |`  B )  ~~> r  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   class class class wbr 4212   dom cdm 4878    |` cres 4880    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989    < clt 9120    <_ cle 9121    - cmin 9291   RR+crp 10612   abscabs 12039    ~~> r crli 12279
This theorem is referenced by:  rlimres2  12355  pnt  21308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-pm 7021  df-rlim 12283
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