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Theorem rlimf 12984
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 12983 . 2  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
2 cnex 9368 . . . 4  |-  CC  e.  _V
3 reex 9378 . . . 4  |-  RR  e.  _V
42, 3elpm2 7249 . . 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 1756    C_ wss 3333   class class class wbr 4297   dom cdm 4845   -->wf 5419  (class class class)co 6096    ^pm cpm 7220   CCcc 9285   RRcr 9286    ~~> r crli 12968
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 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-pm 7222  df-rlim 12972
This theorem is referenced by:  rlimcl  12986  rlimi  12996  rlimclim1  13028  rlimres  13041  rlimmptrcl  13090  rlimo1  13099  o1rlimmul  13101  dvfsumrlim2  21509  rlimcxp  22372
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