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Theorem lo1f 12008
Description: An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
Assertion
Ref Expression
lo1f  |-  ( F  e.  <_ O ( 1 )  ->  F : dom  F --> RR )

Proof of Theorem lo1f
Dummy variables  x  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ello1 12005 . . 3  |-  ( 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 ) )
21simplbi 446 . 2  |-  ( F  e.  <_ O ( 1 )  ->  F  e.  ( RR  ^pm  RR ) )
3 reex 8844 . . . 4  |-  RR  e.  _V
43, 3elpm2 6815 . . 3  |-  ( F  e.  ( RR  ^pm  RR )  <->  ( F : dom  F --> RR  /\  dom  F 
C_  RR ) )
54simplbi 446 . 2  |-  ( F  e.  ( RR  ^pm  RR )  ->  F : dom  F --> RR )
62, 5syl 15 1  |-  ( F  e.  <_ O ( 1 )  ->  F : dom  F --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   class class class wbr 4039   dom cdm 4705   -->wf 5267   ` 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:  lo1res  12049  lo1mptrcl  12111
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791  df-lo1 11981
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