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Theorem ioorval 18945
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 2302 . . 3  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
2 supeq1 7214 . . . 4  |-  ( x  =  A  ->  sup ( x ,  RR* ,  `'  <  )  =  sup ( A ,  RR* ,  `'  <  ) )
3 supeq1 7214 . . . 4  |-  ( x  =  A  ->  sup ( x ,  RR* ,  <  )  =  sup ( A ,  RR* ,  <  ) )
42, 3opeq12d 3820 . . 3  |-  ( x  =  A  ->  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>.  =  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>. )
51, 4ifbieq2d 3598 . 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 4253 . . 3  |-  <. 0 ,  0 >.  e.  _V
8 opex 4253 . . 3  |-  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>.  e.  _V
97, 8ifex 3636 . 2  |-  if ( A  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  ) >.
)  e.  _V
105, 6, 9fvmpt 5618 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 1632    e. wcel 1696   (/)c0 3468   ifcif 3578   <.cop 3656    e. cmpt 4093   `'ccnv 4704   ran crn 4706   ` cfv 5271   supcsup 7209   0cc0 8753   RR*cxr 8882    < clt 8883   (,)cioo 10672
This theorem is referenced by:  ioorinv2  18946  ioorinv  18947  ioorcl  18948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-sup 7210
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