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Theorem iccval 9710
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 9685 . 2  |-  [,]  =  ( y  e.  RR* ,  z  e.  RR*  |->  { x  e.  RR*  |  ( y  <_  x  /\  x  <_  z ) } )
21ixxval 9686 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 1331    e. wcel 1480   {crab 2420   class class class wbr 3929  (class class class)co 5774   RR*cxr 7806    <_ cle 7808   [,]cicc 9681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-icc 9685
This theorem is referenced by:  icc0r  9716  iccmax  9739  fzval2  9800
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