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Theorem rlimres2 12356
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 5192 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  B  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( ( x  e.  B  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
4 rlimres2.2 . . 3  |-  ( ph  ->  ( x  e.  B  |->  C )  ~~> r  D
)
5 rlimres 12353 . . 3  |-  ( ( x  e.  B  |->  C )  ~~> r  D  -> 
( ( x  e.  B  |->  C )  |`  A )  ~~> r  D
)
64, 5syl 16 . 2  |-  ( ph  ->  ( ( x  e.  B  |->  C )  |`  A )  ~~> r  D
)
73, 6eqbrtrrd 4235 1  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    C_ wss 3321   class class class wbr 4213    e. cmpt 4267    |` cres 4881    ~~> r crli 12280
This theorem is referenced by:  divcnv  12634  dvfsumrlimge0  19915  dvfsumrlim2  19917  dfef2  20810  cxp2lim  20816  chtppilimlem2  21169  chpchtlim  21174  pnt2  21308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-pm 7022  df-rlim 12284
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