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Theorem o1dm 12329
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 12325 . . 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 448 . 2  |-  ( F  e.  O ( 1 )  ->  F  e.  ( CC  ^pm  RR ) )
3 cnex 9076 . . . 4  |-  CC  e.  _V
4 reex 9086 . . . 4  |-  RR  e.  _V
53, 4elpm2 7048 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
65simprbi 452 . 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 1726   A.wral 2707   E.wrex 2708    i^i cin 3321    C_ wss 3322   class class class wbr 4215   dom cdm 4881   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^pm cpm 7022   CCcc 8993   RRcr 8994    +oocpnf 9122    <_ cle 9126   [,)cico 10923   abscabs 12044   O (
1 )co1 12285
This theorem is referenced by:  o1bdd  12330  lo1o1  12331  o1lo1  12336  o1lo12  12337  o1co  12385  o1of2  12411  o1rlimmul  12417  o1add2  12422  o1mul2  12423  o1sub2  12424  o1dif  12428  o1cxp  20818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-pm 7024  df-o1 12289
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