MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlimres2 Unicode version

Theorem rlimres2 12031
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 4999 . . 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 12028 . . 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 4046 1  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3153   class class class wbr 4024    e. cmpt 4078    |` cres 4690    ~~> r crli 11955
This theorem is referenced by:  divcnv  12308  dvfsumrlimge0  19373  dvfsumrlim2  19375  dfef2  20261  cxp2lim  20267  chtppilimlem2  20619  chpchtlim  20624  pnt2  20758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-pm 6771  df-rlim 11959
  Copyright terms: Public domain W3C validator