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Theorem ello1 12005
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 4895 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3382 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,)  +oo ) )  =  ( dom  F  i^i  ( x [,)  +oo ) ) )
3 fveq1 5540 . . . . 5  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43breq1d 4049 . . . 4  |-  ( f  =  F  ->  (
( f `  y
)  <_  m  <->  ( F `  y )  <_  m
) )
52, 4raleqbidv 2761 . . 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 2599 . 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 11981 . 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 2938 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164   class class class wbr 4039   dom cdm 4705   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   RRcr 8752    +oocpnf 8880    <_ cle 8884   [,)cico 10674   <_ O ( 1 )clo1 11977
This theorem is referenced by:  ello12  12006  lo1f  12008  lo1dm  12009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-dm 4715  df-iota 5235  df-fv 5279  df-lo1 11981
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