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Theorem iooval2 10736
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 10727 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
2 inrab2 3475 . . . 4  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  (
RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }
3 ressxr 8921 . . . . . 6  |-  RR  C_  RR*
4 sseqin2 3422 . . . . . 6  |-  ( RR  C_  RR*  <->  ( RR*  i^i  RR )  =  RR )
53, 4mpbi 199 . . . . 5  |-  ( RR*  i^i 
RR )  =  RR
6 rabeq 2816 . . . . 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 8 . . . 4  |-  { x  e.  ( RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }  =  {
x  e.  RR  | 
( A  <  x  /\  x  <  B ) }
82, 7eqtri 2336 . . 3  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) }
9 elioore 10733 . . . . . . 7  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR )
109ssriv 3218 . . . . . 6  |-  ( A (,) B )  C_  RR
1110a1i 10 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  C_  RR )
121, 11eqsstr3d 3247 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  C_  RR )
13 df-ss 3200 . . . 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 ) } )
1412, 13sylib 188 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
158, 14syl5reqr 2363 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) } )
161, 15eqtrd 2348 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 358    = wceq 1633    e. wcel 1701   {crab 2581    i^i cin 3185    C_ wss 3186   class class class wbr 4060  (class class class)co 5900   RRcr 8781   RR*cxr 8911    < clt 8912   (,)cioo 10703
This theorem is referenced by:  elioo2  10744  ioomax  10771  ioopos  10773  dfioo2  10791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-pre-lttri 8856  ax-pre-lttrn 8857
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-ioo 10707
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