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Theorem iocval 11569
Description: Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
iocval  |-  ( ( 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 iocval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioc 11537 . 2  |-  (,]  =  ( y  e.  RR* ,  z  e.  RR*  |->  { x  e.  RR*  |  ( y  <  x  /\  x  <_  z ) } )
21ixxval 11540 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 367    = wceq 1398    e. wcel 1823   {crab 2808   class class class wbr 4439  (class class class)co 6270   RR*cxr 9616    < clt 9617    <_ cle 9618   (,]cioc 11533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-xr 9621  df-ioc 11537
This theorem is referenced by:  ioc0  11579  orvclteel  28675
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