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Theorem o1dm 13125
Description: An eventually bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
o1dm  |-  ( F  e.  O(1)  ->  dom  F  C_  RR )

Proof of Theorem o1dm
Dummy variables  x  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elo1 13121 . . 3  |-  ( F  e.  O(1)  <->  ( F  e.  ( CC  ^pm  RR )  /\  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  F  i^i  ( x [,) +oo ) ) ( abs `  ( F `  y
) )  <_  m
) )
21simplbi 460 . 2  |-  ( F  e.  O(1)  ->  F  e.  ( CC  ^pm  RR ) )
3 cnex 9473 . . . 4  |-  CC  e.  _V
4 reex 9483 . . . 4  |-  RR  e.  _V
53, 4elpm2 7353 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
65simprbi 464 . 2  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
72, 6syl 16 1  |-  ( F  e.  O(1)  ->  dom  F  C_  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   A.wral 2798   E.wrex 2799    i^i cin 3434    C_ wss 3435   class class class wbr 4399   dom cdm 4947   -->wf 5521   ` cfv 5525  (class class class)co 6199    ^pm cpm 7324   CCcc 9390   RRcr 9391   +oocpnf 9525    <_ cle 9529   [,)cico 11412   abscabs 12840   O(1)co1 13081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-pm 7326  df-o1 13085
This theorem is referenced by:  o1bdd  13126  lo1o1  13127  o1lo1  13132  o1lo12  13133  o1co  13181  o1of2  13207  o1rlimmul  13213  o1add2  13218  o1mul2  13219  o1sub2  13220  o1dif  13224  o1cxp  22500
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