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Theorem rlimss 12972
Description: Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimss  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )

Proof of Theorem rlimss
StepHypRef Expression
1 rlimpm 12970 . 2  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
2 cnex 9355 . . . 4  |-  CC  e.  _V
3 reex 9365 . . . 4  |-  RR  e.  _V
42, 3elpm2 7236 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
54simprbi 464 . 2  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
61, 5syl 16 1  |-  ( F  ~~> r  A  ->  dom  F 
C_  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756    C_ wss 3323   class class class wbr 4287   dom cdm 4835   -->wf 5409  (class class class)co 6086    ^pm cpm 7207   CCcc 9272   RRcr 9273    ~~> r crli 12955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-pm 7209  df-rlim 12959
This theorem is referenced by:  rlimcl  12973  rlimi  12983  rlimi2  12984  rlimuni  13020  rlimres  13028  rlimeq  13039  rlimcld2  13048  rlimcn1  13058  rlimcn2  13060  rlimo1  13086  o1rlimmul  13088  rlimneg  13116  rlimsqzlem  13118  rlimno1  13123  rlimcxp  22347
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