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Theorem ixxval 10926
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 4217 . . . 4  |-  ( x  =  A  ->  (
x R z  <->  A R
z ) )
21anbi1d 687 . . 3  |-  ( x  =  A  ->  (
( x R z  /\  z S y )  <->  ( A R z  /\  z S y ) ) )
32rabbidv 2950 . 2  |-  ( x  =  A  ->  { z  e.  RR*  |  (
x R z  /\  z S y ) }  =  { z  e. 
RR*  |  ( A R z  /\  z S y ) } )
4 breq2 4218 . . . 4  |-  ( y  =  B  ->  (
z S y  <->  z S B ) )
54anbi2d 686 . . 3  |-  ( y  =  B  ->  (
( A R z  /\  z S y )  <->  ( A R z  /\  z S B ) ) )
65rabbidv 2950 . 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 10611 . . 3  |-  RR*  e.  _V
98rabex 4356 . 2  |-  { z  e.  RR*  |  ( A R z  /\  z S B ) }  e.  _V
103, 6, 7, 9ovmpt2 6211 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 360    = wceq 1653    e. wcel 1726   {crab 2711   class class class wbr 4214  (class class class)co 6083    e. cmpt2 6085   RR*cxr 9121
This theorem is referenced by:  elixx1  10927  ixxin  10935  iooval  10942  iocval  10955  icoval  10956  iccval  10957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-xr 9126
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