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Theorem o1f 13118
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 13115 . . 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 9467 . . . 4  |-  CC  e.  _V
4 reex 9477 . . . 4  |-  RR  e.  _V
53, 4elpm2 7347 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
65simplbi 460 . 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 setvar class
Syntax hints:    -> wi 4    e. wcel 1758   A.wral 2795   E.wrex 2796    i^i cin 3428    C_ wss 3429   class class class wbr 4393   dom cdm 4941   -->wf 5515   ` cfv 5519  (class class class)co 6193    ^pm cpm 7318   CCcc 9384   RRcr 9385   +oocpnf 9519    <_ cle 9523   [,)cico 11406   abscabs 12834   O(1)co1 13075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-pm 7320  df-o1 13079
This theorem is referenced by:  o1res  13149  o1of2  13201  o1rlimmul  13207  o1mptrcl  13211  o1fsum  13387  o1cxp  22494  dchrisum0  22895
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