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Theorem iccval 10955
Description: Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
iccval  |-  ( ( 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 iccval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-icc 10923 . 2  |-  [,]  =  ( y  e.  RR* ,  z  e.  RR*  |->  { x  e.  RR*  |  ( y  <_  x  /\  x  <_  z ) } )
21ixxval 10924 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 359    = wceq 1652    e. wcel 1725   {crab 2709   class class class wbr 4212  (class class class)co 6081   RR*cxr 9119    <_ cle 9121   [,]cicc 10919
This theorem is referenced by:  icc0  10964  iccmax  10986  fzval2  11046  ordtrestixx  17286  cnvordtrestixx  24311  areacirclem5  26296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-xr 9124  df-icc 10923
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