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Theorem iooval2 9727
Description: Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
iooval2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) } )
Distinct variable groups:    x, A    x, B

Proof of Theorem iooval2
StepHypRef Expression
1 iooval 9720 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
2 inrab2 3353 . . . 4  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  (
RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }
3 ressxr 7832 . . . . . 6  |-  RR  C_  RR*
4 sseqin2 3299 . . . . . 6  |-  ( RR  C_  RR*  <->  ( RR*  i^i  RR )  =  RR )
53, 4mpbi 144 . . . . 5  |-  ( RR*  i^i 
RR )  =  RR
6 rabeq 2681 . . . . 5  |-  ( (
RR*  i^i  RR )  =  RR  ->  { x  e.  ( RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }  =  {
x  e.  RR  | 
( A  <  x  /\  x  <  B ) } )
75, 6ax-mp 5 . . . 4  |-  { x  e.  ( RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }  =  {
x  e.  RR  | 
( A  <  x  /\  x  <  B ) }
82, 7eqtri 2161 . . 3  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) }
9 elioore 9724 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR )
109ssriv 3105 . . . . 5  |-  ( A (,) B )  C_  RR
111, 10eqsstrrdi 3154 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  C_  RR )
12 df-ss 3088 . . . 4  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } 
C_  RR  <->  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
1311, 12sylib 121 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
148, 13syl5reqr 2188 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) } )
151, 14eqtrd 2173 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   {crab 2421    i^i cin 3074    C_ wss 3075   class class class wbr 3936  (class class class)co 5781   RRcr 7642   RR*cxr 7822    < clt 7823   (,)cioo 9700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-po 4225  df-iso 4226  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-ioo 9704
This theorem is referenced by:  elioo2  9733  ioomax  9760  ioopos  9762  dfioo2  9786
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