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Theorem ioorval 18929
Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypothesis
Ref Expression
ioorf.1  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. ) )
Assertion
Ref Expression
ioorval  |-  ( A  e.  ran  (,)  ->  ( F `  A )  =  if ( A  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>. ) )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem ioorval
StepHypRef Expression
1 eqeq1 2289 . . 3  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
2 supeq1 7198 . . . 4  |-  ( x  =  A  ->  sup ( x ,  RR* ,  `'  <  )  =  sup ( A ,  RR* ,  `'  <  ) )
3 supeq1 7198 . . . 4  |-  ( x  =  A  ->  sup ( x ,  RR* ,  <  )  =  sup ( A ,  RR* ,  <  ) )
42, 3opeq12d 3804 . . 3  |-  ( x  =  A  ->  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>.  =  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>. )
51, 4ifbieq2d 3585 . 2  |-  ( x  =  A  ->  if ( x  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >. )  =  if ( A  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  ) >.
) )
6 ioorf.1 . 2  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. ) )
7 opex 4237 . . 3  |-  <. 0 ,  0 >.  e.  _V
8 opex 4237 . . 3  |-  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>.  e.  _V
97, 8ifex 3623 . 2  |-  if ( A  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  ) >.
)  e.  _V
105, 6, 9fvmpt 5602 1  |-  ( A  e.  ran  (,)  ->  ( F `  A )  =  if ( A  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   (/)c0 3455   ifcif 3565   <.cop 3643    e. cmpt 4077   `'ccnv 4688   ran crn 4690   ` cfv 5255   supcsup 7193   0cc0 8737   RR*cxr 8866    < clt 8867   (,)cioo 10656
This theorem is referenced by:  ioorinv2  18930  ioorinv  18931  ioorcl  18932
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-iota 5219  df-fun 5257  df-fv 5263  df-sup 7194
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