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Theorem iooval2 9481
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 9474 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
2 inrab2 3288 . . . 4  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  (
RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }
3 ressxr 7628 . . . . . 6  |-  RR  C_  RR*
4 sseqin2 3234 . . . . . 6  |-  ( RR  C_  RR*  <->  ( RR*  i^i  RR )  =  RR )
53, 4mpbi 144 . . . . 5  |-  ( RR*  i^i 
RR )  =  RR
6 rabeq 2625 . . . . 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 7 . . . 4  |-  { x  e.  ( RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }  =  {
x  e.  RR  | 
( A  <  x  /\  x  <  B ) }
82, 7eqtri 2115 . . 3  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) }
9 elioore 9478 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR )
109ssriv 3043 . . . . 5  |-  ( A (,) B )  C_  RR
111, 10syl6eqssr 3092 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  C_  RR )
12 df-ss 3026 . . . 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 2142 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) } )
151, 14eqtrd 2127 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 1296    e. wcel 1445   {crab 2374    i^i cin 3012    C_ wss 3013   class class class wbr 3867  (class class class)co 5690   RRcr 7446   RR*cxr 7618    < clt 7619   (,)cioo 9454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-ioo 9458
This theorem is referenced by:  elioo2  9487  ioomax  9514  ioopos  9516  dfioo2  9540
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