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Theorem elo1 13315
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 5203 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3699 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,) +oo )
)  =  ( dom 
F  i^i  ( x [,) +oo ) ) )
3 fveq1 5865 . . . . . 6  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43fveq2d 5870 . . . . 5  |-  ( f  =  F  ->  ( abs `  ( f `  y ) )  =  ( abs `  ( F `  y )
) )
54breq1d 4457 . . . 4  |-  ( f  =  F  ->  (
( abs `  (
f `  y )
)  <_  m  <->  ( abs `  ( F `  y
) )  <_  m
) )
62, 5raleqbidv 3072 . . 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 2980 . 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 13279 . 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 3263 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475   class class class wbr 4447   dom cdm 4999   ` cfv 5588  (class class class)co 6285    ^pm cpm 7422   CCcc 9491   RRcr 9492   +oocpnf 9626    <_ cle 9630   [,)cico 11532   abscabs 13033   O(1)co1 13275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-dm 5009  df-iota 5551  df-fv 5596  df-o1 13279
This theorem is referenced by:  elo12  13316  o1f  13318  o1dm  13319
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