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Theorem ioorf 21717
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 11618 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5729 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
4 ovelrn 6433 . . . 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 10621 . . . . . . . . 9  |-  0  <_  0
7 df-br 4448 . . . . . . . . 9  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
86, 7mpbi 208 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  <_
9 0xr 9636 . . . . . . . . 9  |-  0  e.  RR*
10 opelxpi 5030 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  0  e.  RR* )  ->  <. 0 ,  0 >.  e.  (
RR*  X.  RR* ) )
119, 9, 10mp2an 672 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  (
RR*  X.  RR* )
12 elin 3687 . . . . . . . 8  |-  ( <.
0 ,  0 >.  e.  (  <_  i^i  ( RR*  X.  RR* ) )  <->  ( <. 0 ,  0 >.  e. 
<_  /\  <. 0 ,  0
>.  e.  ( RR*  X.  RR* ) ) )
138, 11, 12mpbir2an 918 . . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  x  =  ( a (,) b ) )
1615supeq1d 7902 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  `'  <  )  =  sup ( ( a (,) b ) ,  RR* ,  `'  <  ) )
17 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  e.  RR* )
18 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
b  e.  RR* )
19 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  -.  x  =  (/) )
2019neqned 2670 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  x  =/=  (/) )
2115, 20eqnetrrd 2761 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( a (,) b
)  =/=  (/) )
22 df-ioo 11529 . . . . . . . . . . 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 11351 . . . . . . . . . . 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 11351 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  w  e.  RR* )  ->  (
a  <  w  ->  a  <_  w ) )
2722, 23, 24, 25, 26ixxlb 11547 . . . . . . . . . 10  |-  ( ( a  e.  RR*  /\  b  e.  RR*  /\  ( a (,) b )  =/=  (/) )  ->  sup (
( a (,) b
) ,  RR* ,  `'  <  )  =  a )
2817, 18, 21, 27syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( ( a (,) b ) ,  RR* ,  `'  <  )  =  a )
2916, 28eqtrd 2508 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  `'  <  )  =  a )
3015supeq1d 7902 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  sup ( ( a (,) b ) ,  RR* ,  <  ) )
3122, 23, 24, 25, 26ixxub 11546 . . . . . . . . . 10  |-  ( ( a  e.  RR*  /\  b  e.  RR*  /\  ( a (,) b )  =/=  (/) )  ->  sup (
( a (,) b
) ,  RR* ,  <  )  =  b )
3217, 18, 21, 31syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( ( a (,) b ) ,  RR* ,  <  )  =  b )
3330, 32eqtrd 2508 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  b )
3429, 33opeq12d 4221 . . . . . . 7  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >.  =  <. a ,  b
>. )
35 ioon0 11551 . . . . . . . . . . . 12  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
( a (,) b
)  =/=  (/)  <->  a  <  b ) )
3635ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( ( a (,) b )  =/=  (/)  <->  a  <  b ) )
3721, 36mpbid 210 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  <  b )
38 xrltle 11351 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
a  <  b  ->  a  <_  b ) )
3938ad2antrr 725 . . . . . . . . . 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 4448 . . . . . . . . 9  |-  ( a  <_  b  <->  <. a ,  b >.  e.  <_  )
4240, 41sylib 196 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  <_  )
43 opelxpi 5030 . . . . . . . . 9  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  <. a ,  b >.  e.  (
RR*  X.  RR* ) )
4443ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  ( RR*  X.  RR* )
)
4542, 44elind 3688 . . . . . . 7  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
4634, 45eqeltrd 2555 . . . . . 6  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >.  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
4714, 46ifclda 3971 . . . . 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* )
) )
4847ex 434 . . . 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* ) ) ) )
4948rexlimivv 2960 . . 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* ) ) )
505, 49sylbi 195 . 2  |-  ( x  e.  ran  (,)  ->  if ( x  =  (/) , 
<. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. )  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
511, 50fmpti 6042 1  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    i^i cin 3475   (/)c0 3785   ifcif 3939   ~Pcpw 4010   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   ran crn 5000    Fn wfn 5581   -->wf 5582  (class class class)co 6282   supcsup 7896   RRcr 9487   0cc0 9488   RR*cxr 9623    < clt 9624    <_ cle 9625   (,)cioo 11525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-ioo 11529
This theorem is referenced by:  ioorcl  21721  uniioombllem2  21727
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