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Theorem rlimf 12223
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 12222 . 2  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
2 cnex 9005 . . . 4  |-  CC  e.  _V
3 reex 9015 . . . 4  |-  RR  e.  _V
42, 3elpm2 6982 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
54simplbi 447 . 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 set class
Syntax hints:    -> wi 4    e. wcel 1717    C_ wss 3264   class class class wbr 4154   dom cdm 4819   -->wf 5391  (class class class)co 6021    ^pm cpm 6956   CCcc 8922   RRcr 8923    ~~> r crli 12207
This theorem is referenced by:  rlimcl  12225  rlimi  12235  rlimclim1  12267  rlimres  12280  rlimmptrcl  12329  rlimo1  12338  o1rlimmul  12340  dvfsumrlim2  19784  rlimcxp  20680
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-pm 6958  df-rlim 12211
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