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Theorem elo1 12096
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 4961 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3445 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,)  +oo ) )  =  ( dom  F  i^i  ( x [,)  +oo ) ) )
3 fveq1 5607 . . . . . 6  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43fveq2d 5612 . . . . 5  |-  ( f  =  F  ->  ( abs `  ( f `  y ) )  =  ( abs `  ( F `  y )
) )
54breq1d 4114 . . . 4  |-  ( f  =  F  ->  (
( abs `  (
f `  y )
)  <_  m  <->  ( abs `  ( F `  y
) )  <_  m
) )
62, 5raleqbidv 2824 . . 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 2662 . 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 12060 . 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 3001 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 set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620    i^i cin 3227   class class class wbr 4104   dom cdm 4771   ` cfv 5337  (class class class)co 5945    ^pm cpm 6861   CCcc 8825   RRcr 8826    +oocpnf 8954    <_ cle 8958   [,)cico 10750   abscabs 11815   O (
1 )co1 12056
This theorem is referenced by:  elo12  12097  o1f  12099  o1dm  12100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-dm 4781  df-iota 5301  df-fv 5345  df-o1 12060
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