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Theorem rlimres2 12310
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 5150 . . 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 12307 . . 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 4194 1  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    C_ wss 3280   class class class wbr 4172    e. cmpt 4226    |` cres 4839    ~~> r crli 12234
This theorem is referenced by:  divcnv  12588  dvfsumrlimge0  19867  dvfsumrlim2  19869  dfef2  20762  cxp2lim  20768  chtppilimlem2  21121  chpchtlim  21126  pnt2  21260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-pm 6980  df-rlim 12238
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