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Theorem ello1 11989
Description: Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
Assertion
Ref Expression
ello1  |-  ( F  e.  <_ O ( 1 )  <->  ( F  e.  ( RR  ^pm  RR )  /\  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  F  i^i  ( x [,)  +oo ) ) ( F `
 y )  <_  m ) )
Distinct variable group:    x, m, y, F

Proof of Theorem ello1
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 dmeq 4879 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3369 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,)  +oo ) )  =  ( dom  F  i^i  ( x [,)  +oo ) ) )
3 fveq1 5524 . . . . 5  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43breq1d 4033 . . . 4  |-  ( f  =  F  ->  (
( f `  y
)  <_  m  <->  ( F `  y )  <_  m
) )
52, 4raleqbidv 2748 . . 3  |-  ( f  =  F  ->  ( A. y  e.  ( dom  f  i^i  (
x [,)  +oo ) ) ( f `  y
)  <_  m  <->  A. y  e.  ( dom  F  i^i  ( x [,)  +oo ) ) ( F `
 y )  <_  m ) )
652rexbidv 2586 . 2  |-  ( f  =  F  ->  ( E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) 
+oo ) ) ( f `  y )  <_  m  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  F  i^i  ( x [,)  +oo ) ) ( F `
 y )  <_  m ) )
7 df-lo1 11965 . 2  |-  <_ O
( 1 )  =  { f  e.  ( RR  ^pm  RR )  |  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) 
+oo ) ) ( f `  y )  <_  m }
86, 7elrab2 2925 1  |-  ( F  e.  <_ O ( 1 )  <->  ( F  e.  ( RR  ^pm  RR )  /\  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  F  i^i  ( x [,)  +oo ) ) ( F `
 y )  <_  m ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   class class class wbr 4023   dom cdm 4689   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   RRcr 8736    +oocpnf 8864    <_ cle 8868   [,)cico 10658   <_ O ( 1 )clo1 11961
This theorem is referenced by:  ello12  11990  lo1f  11992  lo1dm  11993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-dm 4699  df-iota 5219  df-fv 5263  df-lo1 11965
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