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Theorem iooval2 11445
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 11436 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
2 inrab2 3732 . . . 4  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  (
RR*  i^i  RR )  |  ( A  < 
x  /\  x  <  B ) }
3 ressxr 9539 . . . . . 6  |-  RR  C_  RR*
4 sseqin2 3678 . . . . . 6  |-  ( RR  C_  RR*  <->  ( RR*  i^i  RR )  =  RR )
53, 4mpbi 208 . . . . 5  |-  ( RR*  i^i 
RR )  =  RR
6 rabeq 3072 . . . . 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 2483 . . 3  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  i^i  RR )  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) }
9 elioore 11442 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  x  e.  RR )
109ssriv 3469 . . . . 5  |-  ( A (,) B )  C_  RR
111, 10syl6eqssr 3516 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  C_  RR )
12 df-ss 3451 . . . 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 196 . . 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 2510 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) } )
151, 14eqtrd 2495 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2803    i^i cin 3436    C_ wss 3437   class class class wbr 4401  (class class class)co 6201   RRcr 9393   RR*cxr 9529    < clt 9530   (,)cioo 11412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-pre-lttri 9468  ax-pre-lttrn 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-ioo 11416
This theorem is referenced by:  elioo2  11453  ioomax  11482  ioopos  11484  dfioo2  11508
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