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Theorem ixxssxr 10668
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 5861 . . 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 10666 . . . 4  |-  O :
( RR*  X.  RR* ) --> ~P RR*
4 0elpw 4180 . . . 4  |-  (/)  e.  ~P RR*
53, 4f0cli 5671 . . 3  |-  ( O `
 <. A ,  B >. )  e.  ~P RR*
61, 5eqeltri 2353 . 2  |-  ( A O B )  e. 
~P RR*
7 ovex 5883 . . 3  |-  ( A O B )  e. 
_V
87elpw 3631 . 2  |-  ( ( A O B )  e.  ~P RR*  <->  ( A O B )  C_  RR* )
96, 8mpbi 199 1  |-  ( A O B )  C_  RR*
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ~Pcpw 3625   <.cop 3643   class class class wbr 4023    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   RR*cxr 8866
This theorem is referenced by:  iccssxr  10732  iocssxr  10733  icossxr  10734
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-pow 4188  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-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-xr 8871
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