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Theorem o1f 12323
Description: An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
o1f  |-  ( F  e.  O ( 1 )  ->  F : dom  F --> CC )

Proof of Theorem o1f
Dummy variables  x  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elo1 12320 . . 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 9071 . . . 4  |-  CC  e.  _V
4 reex 9081 . . . 4  |-  RR  e.  _V
53, 4elpm2 7045 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
65simplbi 447 . 2  |-  ( F  e.  ( CC  ^pm  RR )  ->  F : dom  F --> CC )
72, 6syl 16 1  |-  ( F  e.  O ( 1 )  ->  F : dom  F --> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   class class class wbr 4212   dom cdm 4878   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^pm cpm 7019   CCcc 8988   RRcr 8989    +oocpnf 9117    <_ cle 9121   [,)cico 10918   abscabs 12039   O (
1 )co1 12280
This theorem is referenced by:  o1res  12354  o1of2  12406  o1rlimmul  12412  o1mptrcl  12416  o1fsum  12592  o1cxp  20813  dchrisum0  21214
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-pm 7021  df-o1 12284
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