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Theorem elo1 12320
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 5070 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3541 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,)  +oo ) )  =  ( dom  F  i^i  ( x [,)  +oo ) ) )
3 fveq1 5727 . . . . . 6  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43fveq2d 5732 . . . . 5  |-  ( f  =  F  ->  ( abs `  ( f `  y ) )  =  ( abs `  ( F `  y )
) )
54breq1d 4222 . . . 4  |-  ( f  =  F  ->  (
( abs `  (
f `  y )
)  <_  m  <->  ( abs `  ( F `  y
) )  <_  m
) )
62, 5raleqbidv 2916 . . 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 2748 . 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 12284 . 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 3094 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    i^i cin 3319   class class class wbr 4212   dom cdm 4878   ` 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:  elo12  12321  o1f  12323  o1dm  12324
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-dm 4888  df-iota 5418  df-fv 5462  df-o1 12284
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