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Theorem elo1 13568
Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
elo1  |-  ( 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
) )
Distinct variable group:    x, m, y, F

Proof of Theorem elo1
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 dmeq 5055 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3669 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,) +oo )
)  =  ( dom 
F  i^i  ( x [,) +oo ) ) )
3 fveq1 5880 . . . . . 6  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43fveq2d 5885 . . . . 5  |-  ( f  =  F  ->  ( abs `  ( f `  y ) )  =  ( abs `  ( F `  y )
) )
54breq1d 4436 . . . 4  |-  ( f  =  F  ->  (
( abs `  (
f `  y )
)  <_  m  <->  ( abs `  ( F `  y
) )  <_  m
) )
62, 5raleqbidv 3046 . . 3  |-  ( f  =  F  ->  ( A. y  e.  ( dom  f  i^i  (
x [,) +oo )
) ( abs `  (
f `  y )
)  <_  m  <->  A. y  e.  ( dom  F  i^i  ( x [,) +oo ) ) ( abs `  ( F `  y
) )  <_  m
) )
762rexbidv 2953 . 2  |-  ( f  =  F  ->  ( E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  F  i^i  ( x [,) +oo ) ) ( abs `  ( F `  y
) )  <_  m
) )
8 df-o1 13532 . 2  |-  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 }
97, 8elrab2 3237 1  |-  ( 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
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783    i^i cin 3441   class class class wbr 4426   dom cdm 4854   ` cfv 5601  (class class class)co 6305    ^pm cpm 7481   CCcc 9536   RRcr 9537   +oocpnf 9671    <_ cle 9675   [,)cico 11637   abscabs 13276   O(1)co1 13528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-dm 4864  df-iota 5565  df-fv 5609  df-o1 13532
This theorem is referenced by:  elo12  13569  o1f  13571  o1dm  13572
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