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Theorem ioorval 19458
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 2441 . . 3  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
2 supeq1 7442 . . . 4  |-  ( x  =  A  ->  sup ( x ,  RR* ,  `'  <  )  =  sup ( A ,  RR* ,  `'  <  ) )
3 supeq1 7442 . . . 4  |-  ( x  =  A  ->  sup ( x ,  RR* ,  <  )  =  sup ( A ,  RR* ,  <  ) )
42, 3opeq12d 3984 . . 3  |-  ( x  =  A  ->  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>.  =  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>. )
51, 4ifbieq2d 3751 . 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 4419 . . 3  |-  <. 0 ,  0 >.  e.  _V
8 opex 4419 . . 3  |-  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>.  e.  _V
97, 8ifex 3789 . 2  |-  if ( A  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  ) >.
)  e.  _V
105, 6, 9fvmpt 5798 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 1652    e. wcel 1725   (/)c0 3620   ifcif 3731   <.cop 3809    e. cmpt 4258   `'ccnv 4869   ran crn 4871   ` cfv 5446   supcsup 7437   0cc0 8982   RR*cxr 9111    < clt 9112   (,)cioo 10908
This theorem is referenced by:  ioorinv2  19459  ioorinv  19460  ioorcl  19461
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-sup 7438
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