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Theorem o1dm 12316
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 12312 . . 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 447 . 2  |-  ( F  e.  O ( 1 )  ->  F  e.  ( CC  ^pm  RR ) )
3 cnex 9063 . . . 4  |-  CC  e.  _V
4 reex 9073 . . . 4  |-  RR  e.  _V
53, 4elpm2 7037 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
65simprbi 451 . 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 set class
Syntax hints:    -> wi 4    e. wcel 1725   A.wral 2697   E.wrex 2698    i^i cin 3311    C_ wss 3312   class class class wbr 4204   dom cdm 4870   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^pm cpm 7011   CCcc 8980   RRcr 8981    +oocpnf 9109    <_ cle 9113   [,)cico 10910   abscabs 12031   O (
1 )co1 12272
This theorem is referenced by:  o1bdd  12317  lo1o1  12318  o1lo1  12323  o1lo12  12324  o1co  12372  o1of2  12398  o1rlimmul  12404  o1add2  12409  o1mul2  12410  o1sub2  12411  o1dif  12415  o1cxp  20805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-pm 7013  df-o1 12276
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