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Theorem rlimres2 12037
Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
Hypotheses
Ref Expression
rlimres2.1  |-  ( ph  ->  A  C_  B )
rlimres2.2  |-  ( ph  ->  ( x  e.  B  |->  C )  ~~> r  D
)
Assertion
Ref Expression
rlimres2  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    C( x)    D( x)

Proof of Theorem rlimres2
StepHypRef Expression
1 rlimres2.1 . . 3  |-  ( ph  ->  A  C_  B )
2 resmpt 5002 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  B  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
31, 2syl 15 . 2  |-  ( ph  ->  ( ( x  e.  B  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
4 rlimres2.2 . . 3  |-  ( ph  ->  ( x  e.  B  |->  C )  ~~> r  D
)
5 rlimres 12034 . . 3  |-  ( ( x  e.  B  |->  C )  ~~> r  D  -> 
( ( x  e.  B  |->  C )  |`  A )  ~~> r  D
)
64, 5syl 15 . 2  |-  ( ph  ->  ( ( x  e.  B  |->  C )  |`  A )  ~~> r  D
)
73, 6eqbrtrrd 4047 1  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    C_ wss 3154   class class class wbr 4025    e. cmpt 4079    |` cres 4693    ~~> r crli 11961
This theorem is referenced by:  divcnv  12314  dvfsumrlimge0  19379  dvfsumrlim2  19381  dfef2  20267  cxp2lim  20273  chtppilimlem2  20625  chpchtlim  20630  pnt2  20764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-pm 6777  df-rlim 11965
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