MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  o1dm Unicode version

Theorem o1dm 12244
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 12240 . . 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 8997 . . . 4  |-  CC  e.  _V
4 reex 9007 . . . 4  |-  RR  e.  _V
53, 4elpm2 6974 . . 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 1717   A.wral 2642   E.wrex 2643    i^i cin 3255    C_ wss 3256   class class class wbr 4146   dom cdm 4811   -->wf 5383   ` cfv 5387  (class class class)co 6013    ^pm cpm 6948   CCcc 8914   RRcr 8915    +oocpnf 9043    <_ cle 9047   [,)cico 10843   abscabs 11959   O (
1 )co1 12200
This theorem is referenced by:  o1bdd  12245  lo1o1  12246  o1lo1  12251  o1lo12  12252  o1co  12300  o1of2  12326  o1rlimmul  12332  o1add2  12337  o1mul2  12338  o1sub2  12339  o1dif  12343  o1cxp  20673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-pm 6950  df-o1 12204
  Copyright terms: Public domain W3C validator