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Theorem rlimss 13407
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 13405 . 2  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
2 cnex 9562 . . . 4  |-  CC  e.  _V
3 reex 9572 . . . 4  |-  RR  e.  _V
42, 3elpm2 7443 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
54simprbi 462 . 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 1823    C_ wss 3461   class class class wbr 4439   dom cdm 4988   -->wf 5566  (class class class)co 6270    ^pm cpm 7413   CCcc 9479   RRcr 9480    ~~> r crli 13390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-pm 7415  df-rlim 13394
This theorem is referenced by:  rlimcl  13408  rlimi  13418  rlimi2  13419  rlimuni  13455  rlimres  13463  rlimeq  13474  rlimcld2  13483  rlimcn1  13493  rlimcn2  13495  rlimo1  13521  o1rlimmul  13523  rlimneg  13551  rlimsqzlem  13553  rlimno1  13558  rlimcxp  23501
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