MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ello1 Unicode version

Theorem ello1 12272
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 5037 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3509 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,)  +oo ) )  =  ( dom  F  i^i  ( x [,)  +oo ) ) )
3 fveq1 5694 . . . . 5  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43breq1d 4190 . . . 4  |-  ( f  =  F  ->  (
( f `  y
)  <_  m  <->  ( F `  y )  <_  m
) )
52, 4raleqbidv 2884 . . 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 2717 . 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 12248 . 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 3062 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675    i^i cin 3287   class class class wbr 4180   dom cdm 4845   ` cfv 5421  (class class class)co 6048    ^pm cpm 6986   RRcr 8953    +oocpnf 9081    <_ cle 9085   [,)cico 10882   <_ O ( 1 )clo1 12244
This theorem is referenced by:  ello12  12273  lo1f  12275  lo1dm  12276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-dm 4855  df-iota 5385  df-fv 5429  df-lo1 12248
  Copyright terms: Public domain W3C validator