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Theorem ello1 13350
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 5213 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3695 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,) +oo )
)  =  ( dom 
F  i^i  ( x [,) +oo ) ) )
3 fveq1 5871 . . . . 5  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43breq1d 4466 . . . 4  |-  ( f  =  F  ->  (
( f `  y
)  <_  m  <->  ( F `  y )  <_  m
) )
52, 4raleqbidv 3068 . . 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 2975 . 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 13326 . 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 3259 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    i^i cin 3470   class class class wbr 4456   dom cdm 5008   ` cfv 5594  (class class class)co 6296    ^pm cpm 7439   RRcr 9508   +oocpnf 9642    <_ cle 9646   [,)cico 11556   <_O(1)clo1 13322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-dm 5018  df-iota 5557  df-fv 5602  df-lo1 13326
This theorem is referenced by:  ello12  13351  lo1f  13353  lo1dm  13354
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