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Theorem ixxval 11506
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 4395 . . . 4  |-  ( x  =  A  ->  (
x R z  <->  A R
z ) )
21anbi1d 703 . . 3  |-  ( x  =  A  ->  (
( x R z  /\  z S y )  <->  ( A R z  /\  z S y ) ) )
32rabbidv 3048 . 2  |-  ( x  =  A  ->  { z  e.  RR*  |  (
x R z  /\  z S y ) }  =  { z  e. 
RR*  |  ( A R z  /\  z S y ) } )
4 breq2 4396 . . . 4  |-  ( y  =  B  ->  (
z S y  <->  z S B ) )
54anbi2d 702 . . 3  |-  ( y  =  B  ->  (
( A R z  /\  z S y )  <->  ( A R z  /\  z S B ) ) )
65rabbidv 3048 . 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 11178 . . 3  |-  RR*  e.  _V
98rabex 4542 . 2  |-  { z  e.  RR*  |  ( A R z  /\  z S B ) }  e.  _V
103, 6, 7, 9ovmpt2 6373 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  =  { z  e.  RR*  |  ( A R z  /\  z S B ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   {crab 2755   class class class wbr 4392  (class class class)co 6232    |-> cmpt2 6234   RR*cxr 9575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5487  df-fun 5525  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-xr 9580
This theorem is referenced by:  elixx1  11507  ixxin  11515  iooval  11522  iocval  11535  icoval  11536  iccval  11537
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