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

Theorem rlimres2 12051
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 5016 . . 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 12048 . . 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 4061 1  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165   class class class wbr 4039    e. cmpt 4093    |` cres 4707    ~~> r crli 11975
This theorem is referenced by:  divcnv  12328  dvfsumrlimge0  19393  dvfsumrlim2  19395  dfef2  20281  cxp2lim  20287  chtppilimlem2  20639  chpchtlim  20644  pnt2  20778
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-mpt 4095  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
  Copyright terms: Public domain W3C validator