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Theorem ioorf 22604
Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)
Hypothesis
Ref Expression
ioorf.1  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <.inf ( x ,  RR* ,  <  ) ,  sup ( x , 
RR* ,  <  ) >.
) )
Assertion
Ref Expression
ioorf  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)

Proof of Theorem ioorf
Dummy variables  a 
b  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioorf.1 . 2  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <.inf ( x ,  RR* ,  <  ) ,  sup ( x , 
RR* ,  <  ) >.
) )
2 ioof 11757 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5739 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
4 ovelrn 6464 . . . 4  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( x  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  x  =  ( a (,) b ) ) )
52, 3, 4mp2b 10 . . 3  |-  ( x  e.  ran  (,)  <->  E. a  e.  RR*  E. b  e. 
RR*  x  =  ( a (,) b ) )
6 0le0 10721 . . . . . . . . 9  |-  0  <_  0
7 df-br 4396 . . . . . . . . 9  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
86, 7mpbi 213 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  <_
9 0xr 9705 . . . . . . . . 9  |-  0  e.  RR*
10 opelxpi 4871 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  0  e.  RR* )  ->  <. 0 ,  0 >.  e.  (
RR*  X.  RR* ) )
119, 9, 10mp2an 686 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  (
RR*  X.  RR* )
12 elin 3608 . . . . . . . 8  |-  ( <.
0 ,  0 >.  e.  (  <_  i^i  ( RR*  X.  RR* ) )  <->  ( <. 0 ,  0 >.  e. 
<_  /\  <. 0 ,  0
>.  e.  ( RR*  X.  RR* ) ) )
138, 11, 12mpbir2an 934 . . . . . . 7  |-  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR*  X. 
RR* ) )
1413a1i 11 . . . . . 6  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  x  =  (/) )  ->  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR*  X. 
RR* ) ) )
15 simplr 770 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  x  =  ( a (,) b ) )
1615infeq1d 8011 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> inf ( x ,  RR* ,  <  )  = inf (
( a (,) b
) ,  RR* ,  <  ) )
17 simplll 776 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  e.  RR* )
18 simpllr 777 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
b  e.  RR* )
19 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  -.  x  =  (/) )
2019neqned 2650 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  x  =/=  (/) )
2115, 20eqnetrrd 2711 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( a (,) b
)  =/=  (/) )
22 df-ioo 11664 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
23 idd 24 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  b  e.  RR* )  ->  (
w  <  b  ->  w  <  b ) )
24 xrltle 11471 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  b  e.  RR* )  ->  (
w  <  b  ->  w  <_  b ) )
25 idd 24 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  w  e.  RR* )  ->  (
a  <  w  ->  a  <  w ) )
26 xrltle 11471 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  w  e.  RR* )  ->  (
a  <  w  ->  a  <_  w ) )
2722, 23, 24, 25, 26ixxlb 11682 . . . . . . . . . 10  |-  ( ( a  e.  RR*  /\  b  e.  RR*  /\  ( a (,) b )  =/=  (/) )  -> inf ( ( a (,) b ) ,  RR* ,  <  )  =  a )
2817, 18, 21, 27syl3anc 1292 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> inf ( ( a (,) b ) ,  RR* ,  <  )  =  a )
2916, 28eqtrd 2505 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> inf ( x ,  RR* ,  <  )  =  a )
3015supeq1d 7978 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  sup ( ( a (,) b ) ,  RR* ,  <  ) )
3122, 23, 24, 25, 26ixxub 11681 . . . . . . . . . 10  |-  ( ( a  e.  RR*  /\  b  e.  RR*  /\  ( a (,) b )  =/=  (/) )  ->  sup (
( a (,) b
) ,  RR* ,  <  )  =  b )
3217, 18, 21, 31syl3anc 1292 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( ( a (,) b ) ,  RR* ,  <  )  =  b )
3330, 32eqtrd 2505 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  b )
3429, 33opeq12d 4166 . . . . . . 7  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <.inf ( x ,  RR* ,  <  ) ,  sup ( x ,  RR* ,  <  ) >.  =  <. a ,  b >. )
35 ioon0 11687 . . . . . . . . . . . 12  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
( a (,) b
)  =/=  (/)  <->  a  <  b ) )
3635ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( ( a (,) b )  =/=  (/)  <->  a  <  b ) )
3721, 36mpbid 215 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  <  b )
38 xrltle 11471 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
a  <  b  ->  a  <_  b ) )
3938ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( a  <  b  ->  a  <_  b )
)
4037, 39mpd 15 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  <_  b )
41 df-br 4396 . . . . . . . . 9  |-  ( a  <_  b  <->  <. a ,  b >.  e.  <_  )
4240, 41sylib 201 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  <_  )
43 opelxpi 4871 . . . . . . . . 9  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  <. a ,  b >.  e.  (
RR*  X.  RR* ) )
4443ad2antrr 740 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  ( RR*  X.  RR* )
)
4542, 44elind 3609 . . . . . . 7  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
4634, 45eqeltrd 2549 . . . . . 6  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <.inf ( x ,  RR* ,  <  ) ,  sup ( x ,  RR* ,  <  ) >.  e.  (  <_  i^i  ( RR*  X. 
RR* ) ) )
4714, 46ifclda 3904 . . . . 5  |-  ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  x  =  (
a (,) b ) )  ->  if (
x  =  (/) ,  <. 0 ,  0 >. , 
<.inf ( x ,  RR* ,  <  ) ,  sup ( x ,  RR* ,  <  ) >. )  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
4847ex 441 . . . 4  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
x  =  ( a (,) b )  ->  if ( x  =  (/) , 
<. 0 ,  0
>. ,  <.inf ( x ,  RR* ,  <  ) ,  sup ( x , 
RR* ,  <  ) >.
)  e.  (  <_  i^i  ( RR*  X.  RR* )
) ) )
4948rexlimivv 2876 . . 3  |-  ( E. a  e.  RR*  E. b  e.  RR*  x  =  ( a (,) b )  ->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <.inf ( x ,  RR* ,  <  ) ,  sup ( x ,  RR* ,  <  ) >. )  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
505, 49sylbi 200 . 2  |-  ( x  e.  ran  (,)  ->  if ( x  =  (/) , 
<. 0 ,  0
>. ,  <.inf ( x ,  RR* ,  <  ) ,  sup ( x , 
RR* ,  <  ) >.
)  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
511, 50fmpti 6060 1  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757    i^i cin 3389   (/)c0 3722   ifcif 3872   ~Pcpw 3942   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   ran crn 4840    Fn wfn 5584   -->wf 5585  (class class class)co 6308   supcsup 7972  infcinf 7973   RRcr 9556   0cc0 9557   RR*cxr 9692    < clt 9693    <_ cle 9694   (,)cioo 11660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-ioo 11664
This theorem is referenced by:  ioorcl  22608  uniioombllem2  22619
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