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Theorem ixxval 11528
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 4445 . . . 4  |-  ( x  =  A  ->  (
x R z  <->  A R
z ) )
21anbi1d 704 . . 3  |-  ( x  =  A  ->  (
( x R z  /\  z S y )  <->  ( A R z  /\  z S y ) ) )
32rabbidv 3100 . 2  |-  ( x  =  A  ->  { z  e.  RR*  |  (
x R z  /\  z S y ) }  =  { z  e. 
RR*  |  ( A R z  /\  z S y ) } )
4 breq2 4446 . . . 4  |-  ( y  =  B  ->  (
z S y  <->  z S B ) )
54anbi2d 703 . . 3  |-  ( y  =  B  ->  (
( A R z  /\  z S y )  <->  ( A R z  /\  z S B ) ) )
65rabbidv 3100 . 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 11208 . . 3  |-  RR*  e.  _V
98rabex 4593 . 2  |-  { z  e.  RR*  |  ( A R z  /\  z S B ) }  e.  _V
103, 6, 7, 9ovmpt2 6415 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 369    = wceq 1374    e. wcel 1762   {crab 2813   class class class wbr 4442  (class class class)co 6277    |-> cmpt2 6279   RR*cxr 9618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-xr 9623
This theorem is referenced by:  elixx1  11529  ixxin  11537  iooval  11544  iocval  11557  icoval  11558  iccval  11559
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