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Theorem ioorf 18928
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
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
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. ) )
2 ioof 10741 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5389 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
4 ovelrn 5996 . . . 4  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( x  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  x  =  ( a (,) b ) ) )
52, 3, 4mp2b 9 . . 3  |-  ( x  e.  ran  (,)  <->  E. a  e.  RR*  E. b  e. 
RR*  x  =  ( a (,) b ) )
6 0le0 9827 . . . . . . . . 9  |-  0  <_  0
7 df-br 4024 . . . . . . . . 9  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
86, 7mpbi 199 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  <_
9 0xr 8878 . . . . . . . . 9  |-  0  e.  RR*
10 opelxpi 4721 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  0  e.  RR* )  ->  <. 0 ,  0 >.  e.  (
RR*  X.  RR* ) )
119, 9, 10mp2an 653 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  (
RR*  X.  RR* )
12 elin 3358 . . . . . . . 8  |-  ( <.
0 ,  0 >.  e.  (  <_  i^i  ( RR*  X.  RR* ) )  <->  ( <. 0 ,  0 >.  e. 
<_  /\  <. 0 ,  0
>.  e.  ( RR*  X.  RR* ) ) )
138, 11, 12mpbir2an 886 . . . . . . 7  |-  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR*  X. 
RR* ) )
1413a1i 10 . . . . . 6  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  x  =  (/) )  ->  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR*  X. 
RR* ) ) )
15 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  x  =  ( a (,) b ) )
1615supeq1d 7199 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  `'  <  )  =  sup ( ( a (,) b ) ,  RR* ,  `'  <  ) )
17 simplll 734 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  e.  RR* )
18 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
b  e.  RR* )
19 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  -.  x  =  (/) )
20 df-ne 2448 . . . . . . . . . . . 12  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
2119, 20sylibr 203 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  x  =/=  (/) )
2215, 21eqnetrrd 2466 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( a (,) b
)  =/=  (/) )
23 df-ioo 10660 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
24 idd 21 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  b  e.  RR* )  ->  (
w  <  b  ->  w  <  b ) )
25 xrltle 10483 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  b  e.  RR* )  ->  (
w  <  b  ->  w  <_  b ) )
26 idd 21 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  w  e.  RR* )  ->  (
a  <  w  ->  a  <  w ) )
27 xrltle 10483 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  w  e.  RR* )  ->  (
a  <  w  ->  a  <_  w ) )
2823, 24, 25, 26, 27ixxlb 10678 . . . . . . . . . 10  |-  ( ( a  e.  RR*  /\  b  e.  RR*  /\  ( a (,) b )  =/=  (/) )  ->  sup (
( a (,) b
) ,  RR* ,  `'  <  )  =  a )
2917, 18, 22, 28syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( ( a (,) b ) ,  RR* ,  `'  <  )  =  a )
3016, 29eqtrd 2315 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  `'  <  )  =  a )
3115supeq1d 7199 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  sup ( ( a (,) b ) ,  RR* ,  <  ) )
3223, 24, 25, 26, 27ixxub 10677 . . . . . . . . . 10  |-  ( ( a  e.  RR*  /\  b  e.  RR*  /\  ( a (,) b )  =/=  (/) )  ->  sup (
( a (,) b
) ,  RR* ,  <  )  =  b )
3317, 18, 22, 32syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( ( a (,) b ) ,  RR* ,  <  )  =  b )
3431, 33eqtrd 2315 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  b )
3530, 34opeq12d 3804 . . . . . . 7  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >.  =  <. a ,  b
>. )
36 ioon0 10682 . . . . . . . . . . . 12  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
( a (,) b
)  =/=  (/)  <->  a  <  b ) )
3736ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( ( a (,) b )  =/=  (/)  <->  a  <  b ) )
3822, 37mpbid 201 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  <  b )
39 xrltle 10483 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
a  <  b  ->  a  <_  b ) )
4039ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( a  <  b  ->  a  <_  b )
)
4138, 40mpd 14 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  <_  b )
42 df-br 4024 . . . . . . . . 9  |-  ( a  <_  b  <->  <. a ,  b >.  e.  <_  )
4341, 42sylib 188 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  <_  )
44 opelxpi 4721 . . . . . . . . 9  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  <. a ,  b >.  e.  (
RR*  X.  RR* ) )
4544ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  ( RR*  X.  RR* )
)
46 elin 3358 . . . . . . . 8  |-  ( <.
a ,  b >.  e.  (  <_  i^i  ( RR*  X.  RR* ) )  <->  ( <. a ,  b >.  e.  <_  /\ 
<. a ,  b >.  e.  ( RR*  X.  RR* )
) )
4743, 45, 46sylanbrc 645 . . . . . . 7  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
4835, 47eqeltrd 2357 . . . . . 6  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >.  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
4914, 48ifclda 3592 . . . . 5  |-  ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  x  =  (
a (,) b ) )  ->  if (
x  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >. )  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
5049ex 423 . . . 4  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
x  =  ( a (,) b )  ->  if ( x  =  (/) , 
<. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. )  e.  (  <_  i^i  ( RR*  X.  RR* ) ) ) )
5150rexlimivv 2672 . . 3  |-  ( E. a  e.  RR*  E. b  e.  RR*  x  =  ( a (,) b )  ->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >. )  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
525, 51sylbi 187 . 2  |-  ( x  e.  ran  (,)  ->  if ( x  =  (/) , 
<. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. )  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
531, 52fmpti 5683 1  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    i^i cin 3151   (/)c0 3455   ifcif 3565   ~Pcpw 3625   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   ran crn 4690    Fn wfn 5250   -->wf 5251  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   RR*cxr 8866    < clt 8867    <_ cle 8868   (,)cioo 10656
This theorem is referenced by:  ioorcl  18932  uniioombllem2  18938
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  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-ioo 10660
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