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Theorem ixxval 10664
Description: Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
Hypothesis
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
Assertion
Ref Expression
ixxval  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  { z  e.  RR*  |  ( A R z  /\  z S B ) } )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, R, y, z    x, S, y, z
Allowed substitution hints:    O( x, y, z)

Proof of Theorem ixxval
StepHypRef Expression
1 breq1 4026 . . . 4  |-  ( x  =  A  ->  (
x R z  <->  A R
z ) )
21anbi1d 685 . . 3  |-  ( x  =  A  ->  (
( x R z  /\  z S y )  <->  ( A R z  /\  z S y ) ) )
32rabbidv 2780 . 2  |-  ( x  =  A  ->  { z  e.  RR*  |  (
x R z  /\  z S y ) }  =  { z  e. 
RR*  |  ( A R z  /\  z S y ) } )
4 breq2 4027 . . . 4  |-  ( y  =  B  ->  (
z S y  <->  z S B ) )
54anbi2d 684 . . 3  |-  ( y  =  B  ->  (
( A R z  /\  z S y )  <->  ( A R z  /\  z S B ) ) )
65rabbidv 2780 . 2  |-  ( y  =  B  ->  { z  e.  RR*  |  ( A R z  /\  z S y ) }  =  { z  e. 
RR*  |  ( A R z  /\  z S B ) } )
7 ixx.1 . 2  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
8 xrex 10351 . . 3  |-  RR*  e.  _V
98rabex 4165 . 2  |-  { z  e.  RR*  |  ( A R z  /\  z S B ) }  e.  _V
103, 6, 7, 9ovmpt2 5983 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  { z  e.  RR*  |  ( A R z  /\  z S B ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   class class class wbr 4023  (class class class)co 5858    e. cmpt2 5860   RR*cxr 8866
This theorem is referenced by:  elixx1  10665  ixxin  10673  iooval  10680  iocval  10693  icoval  10694  iccval  10695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-xr 8871
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