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Theorem o1f 13434
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 13431 . . 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 458 . 2  |-  ( F  e.  O(1)  ->  F  e.  ( CC  ^pm  RR ) )
3 cnex 9562 . . . 4  |-  CC  e.  _V
4 reex 9572 . . . 4  |-  RR  e.  _V
53, 4elpm2 7443 . . 3  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
65simplbi 458 . 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 1823   A.wral 2804   E.wrex 2805    i^i cin 3460    C_ wss 3461   class class class wbr 4439   dom cdm 4988   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^pm cpm 7413   CCcc 9479   RRcr 9480   +oocpnf 9614    <_ cle 9618   [,)cico 11534   abscabs 13149   O(1)co1 13391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-pm 7415  df-o1 13395
This theorem is referenced by:  o1res  13465  o1of2  13517  o1rlimmul  13523  o1mptrcl  13527  o1fsum  13709  o1cxp  23502  dchrisum0  23903
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