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Theorem ioorf 19026
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 10830 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5469 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
4 ovelrn 6080 . . . 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 9914 . . . . . . . . 9  |-  0  <_  0
7 df-br 4103 . . . . . . . . 9  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
86, 7mpbi 199 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  <_
9 0xr 8965 . . . . . . . . 9  |-  0  e.  RR*
10 opelxpi 4800 . . . . . . . . 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 3434 . . . . . . . 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 7286 . . . . . . . . 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 2523 . . . . . . . . . . . 12  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
2119, 20sylibr 203 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  x  =/=  (/) )
2215, 21eqnetrrd 2541 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( a (,) b
)  =/=  (/) )
23 df-ioo 10749 . . . . . . . . . . 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 10572 . . . . . . . . . . 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 10572 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  w  e.  RR* )  ->  (
a  <  w  ->  a  <_  w ) )
2823, 24, 25, 26, 27ixxlb 10767 . . . . . . . . . 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 2390 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  `'  <  )  =  a )
3115supeq1d 7286 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  sup ( ( a (,) b ) ,  RR* ,  <  ) )
3223, 24, 25, 26, 27ixxub 10766 . . . . . . . . . 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 2390 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  b )
3530, 34opeq12d 3883 . . . . . . 7  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >.  =  <. a ,  b
>. )
36 ioon0 10771 . . . . . . . . . . . 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 10572 . . . . . . . . . . 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 4103 . . . . . . . . 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 4800 . . . . . . . . 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 3434 . . . . . . . 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 2432 . . . . . 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 3668 . . . . 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 2748 . . 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 5763 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 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620    i^i cin 3227   (/)c0 3531   ifcif 3641   ~Pcpw 3701   <.cop 3719   class class class wbr 4102    e. cmpt 4156    X. cxp 4766   `'ccnv 4767   ran crn 4769    Fn wfn 5329   -->wf 5330  (class class class)co 5942   supcsup 7280   RRcr 8823   0cc0 8824   RR*cxr 8953    < clt 8954    <_ cle 8955   (,)cioo 10745
This theorem is referenced by:  ioorcl  19030  uniioombllem2  19036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-n0 10055  df-z 10114  df-uz 10320  df-q 10406  df-ioo 10749
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