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Theorem ello1 12314
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 5073 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3543 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,)  +oo ) )  =  ( dom  F  i^i  ( x [,)  +oo ) ) )
3 fveq1 5730 . . . . 5  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43breq1d 4225 . . . 4  |-  ( f  =  F  ->  (
( f `  y
)  <_  m  <->  ( F `  y )  <_  m
) )
52, 4raleqbidv 2918 . . 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 2750 . 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 12290 . 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 3096 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    i^i cin 3321   class class class wbr 4215   dom cdm 4881   ` cfv 5457  (class class class)co 6084    ^pm cpm 7022   RRcr 8994    +oocpnf 9122    <_ cle 9126   [,)cico 10923   <_ O ( 1 )clo1 12286
This theorem is referenced by:  ello12  12315  lo1f  12317  lo1dm  12318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-dm 4891  df-iota 5421  df-fv 5465  df-lo1 12290
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