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Theorem rlimf 13304
Description: Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimf  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )

Proof of Theorem rlimf
StepHypRef Expression
1 rlimpm 13303 . 2  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
2 cnex 9585 . . . 4  |-  CC  e.  _V
3 reex 9595 . . . 4  |-  RR  e.  _V
42, 3elpm2 7462 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
54simplbi 460 . 2  |-  ( F  e.  ( CC  ^pm  RR )  ->  F : dom  F --> CC )
61, 5syl 16 1  |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767    C_ wss 3481   class class class wbr 4453   dom cdm 5005   -->wf 5590  (class class class)co 6295    ^pm cpm 7433   CCcc 9502   RRcr 9503    ~~> r crli 13288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-pm 7435  df-rlim 13292
This theorem is referenced by:  rlimcl  13306  rlimi  13316  rlimclim1  13348  rlimres  13361  rlimmptrcl  13410  rlimo1  13419  o1rlimmul  13421  dvfsumrlim2  22301  rlimcxp  23169
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