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Theorem iooval2 9947
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 9940 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
2 elioore 9944 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR )
32ssriv 3174 . . . . 5  |-  ( A (,) B )  C_  RR
41, 3eqsstrrdi 3223 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  C_  RR )
5 df-ss 3157 . . . 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 ) } )
64, 5sylib 122 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
7 inrab2 3423 . . . 4  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  (
RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }
8 ressxr 8032 . . . . . 6  |-  RR  C_  RR*
9 sseqin2 3369 . . . . . 6  |-  ( RR  C_  RR*  <->  ( RR*  i^i  RR )  =  RR )
108, 9mpbi 145 . . . . 5  |-  ( RR*  i^i 
RR )  =  RR
11 rabeq 2744 . . . . 5  |-  ( (
RR*  i^i  RR )  =  RR  ->  { x  e.  ( RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }  =  {
x  e.  RR  | 
( A  <  x  /\  x  <  B ) } )
1210, 11ax-mp 5 . . . 4  |-  { x  e.  ( RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }  =  {
x  e.  RR  | 
( A  <  x  /\  x  <  B ) }
137, 12eqtri 2210 . . 3  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) }
146, 13eqtr3di 2237 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) } )
151, 14eqtrd 2222 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 104    = wceq 1364    e. wcel 2160   {crab 2472    i^i cin 3143    C_ wss 3144   class class class wbr 4018  (class class class)co 5897   RRcr 7841   RR*cxr 8022    < clt 8023   (,)cioo 9920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-ioo 9924
This theorem is referenced by:  elioo2  9953  ioomax  9980  ioopos  9982  dfioo2  10006
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