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Theorem o1dm 13000
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 12996 . . 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 9355 . . . 4  |-  CC  e.  _V
4 reex 9365 . . . 4  |-  RR  e.  _V
53, 4elpm2 7236 . . 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 1756   A.wral 2710   E.wrex 2711    i^i cin 3322    C_ wss 3323   class class class wbr 4287   dom cdm 4835   -->wf 5409   ` cfv 5413  (class class class)co 6086    ^pm cpm 7207   CCcc 9272   RRcr 9273   +oocpnf 9407    <_ cle 9411   [,)cico 11294   abscabs 12715   O(1)co1 12956
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-pm 7209  df-o1 12960
This theorem is referenced by:  o1bdd  13001  lo1o1  13002  o1lo1  13007  o1lo12  13008  o1co  13056  o1of2  13082  o1rlimmul  13088  o1add2  13093  o1mul2  13094  o1sub2  13095  o1dif  13099  o1cxp  22348
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