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Theorem ixxssxr 10928
Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
Hypothesis
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
Assertion
Ref Expression
ixxssxr  |-  ( A O B )  C_  RR*
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 ixxssxr
StepHypRef Expression
1 df-ov 6084 . . 3  |-  ( A O B )  =  ( O `  <. A ,  B >. )
2 ixx.1 . . . . 5  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
32ixxf 10926 . . . 4  |-  O :
( RR*  X.  RR* ) --> ~P RR*
4 0elpw 4369 . . . 4  |-  (/)  e.  ~P RR*
53, 4f0cli 5880 . . 3  |-  ( O `
 <. A ,  B >. )  e.  ~P RR*
61, 5eqeltri 2506 . 2  |-  ( A O B )  e. 
~P RR*
7 ovex 6106 . . 3  |-  ( A O B )  e. 
_V
87elpw 3805 . 2  |-  ( ( A O B )  e.  ~P RR*  <->  ( A O B )  C_  RR* )
96, 8mpbi 200 1  |-  ( A O B )  C_  RR*
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709    C_ wss 3320   ~Pcpw 3799   <.cop 3817   class class class wbr 4212    X. cxp 4876   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   RR*cxr 9119
This theorem is referenced by:  iccssxr  10993  iocssxr  10994  icossxr  10995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-xr 9124
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