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Theorem rlimres 12048
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 3402 . . . . . . . 8  |-  ( dom 
F  i^i  B )  C_ 
dom  F
2 ssralv 3250 . . . . . . . 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 2663 . . . . . 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 2631 . . . . 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 552 . . . 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 10 . . 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 11991 . . . 4  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
9 rlimss 11992 . . . 4  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
10 eqidd 2297 . . . 4  |-  ( ( F  ~~> r  A  /\  z  e.  dom  F )  ->  ( F `  z )  =  ( F `  z ) )
118, 9, 10rlim 11985 . . 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 5424 . . . . . 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 643 . . . . 5  |-  ( F  ~~> r  A  ->  ( F  |`  ( dom  F  i^i  B ) ) : ( dom  F  i^i  B ) --> CC )
14 resres 4984 . . . . . . 7  |-  ( ( F  |`  dom  F )  |`  B )  =  ( F  |`  ( dom  F  i^i  B ) )
15 ffn 5405 . . . . . . . . 9  |-  ( F : dom  F --> CC  ->  F  Fn  dom  F )
16 fnresdm 5369 . . . . . . . . 9  |-  ( F  Fn  dom  F  -> 
( F  |`  dom  F
)  =  F )
178, 15, 163syl 18 . . . . . . . 8  |-  ( F  ~~> r  A  ->  ( F  |`  dom  F )  =  F )
1817reseq1d 4970 . . . . . . 7  |-  ( F  ~~> r  A  ->  (
( F  |`  dom  F
)  |`  B )  =  ( F  |`  B ) )
1914, 18syl5eqr 2342 . . . . . 6  |-  ( F  ~~> r  A  ->  ( F  |`  ( dom  F  i^i  B ) )  =  ( F  |`  B ) )
2019feq1d 5395 . . . . 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 201 . . . 4  |-  ( F  ~~> r  A  ->  ( F  |`  B ) : ( dom  F  i^i  B ) --> CC )
221, 9syl5ss 3203 . . . 4  |-  ( F  ~~> r  A  ->  ( dom  F  i^i  B ) 
C_  RR )
23 inss2 3403 . . . . . . 7  |-  ( dom 
F  i^i  B )  C_  B
2423sseli 3189 . . . . . 6  |-  ( z  e.  ( dom  F  i^i  B )  ->  z  e.  B )
25 fvres 5558 . . . . . 6  |-  ( z  e.  B  ->  (
( F  |`  B ) `
 z )  =  ( F `  z
) )
2624, 25syl 15 . . . . 5  |-  ( z  e.  ( dom  F  i^i  B )  ->  (
( F  |`  B ) `
 z )  =  ( F `  z
) )
2726adantl 452 . . . 4  |-  ( ( F  ~~> r  A  /\  z  e.  ( dom  F  i^i  B ) )  ->  ( ( F  |`  B ) `  z
)  =  ( F `
 z ) )
2821, 22, 27rlim 11985 . . 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 259 . 2  |-  ( F  ~~> r  A  ->  ( F 
~~> r  A  ->  ( F  |`  B )  ~~> r  A
) )
3029pm2.43i 43 1  |-  ( F  ~~> r  A  ->  ( F  |`  B )  ~~> r  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   class class class wbr 4039   dom cdm 4705    |` cres 4707    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752    < clt 8883    <_ cle 8884    - cmin 9053   RR+crp 10370   abscabs 11735    ~~> r crli 11975
This theorem is referenced by:  rlimres2  12051  pnt  20779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791  df-rlim 11979
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