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Theorem ioorinv 19468
Description: The function  F is an "inverse" of sorts to the open interval function. (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
ioorinv  |-  ( A  e.  ran  (,)  ->  ( (,) `  ( F `
 A ) )  =  A )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem ioorinv
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioof 11002 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 ffn 5591 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3 ovelrn 6222 . . . 4  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( A  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b ) ) )
41, 2, 3mp2b 10 . . 3  |-  ( A  e.  ran  (,)  <->  E. a  e.  RR*  E. b  e. 
RR*  A  =  (
a (,) b ) )
5 ioorf.1 . . . . . . . . 9  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. ) )
65ioorinv2 19467 . . . . . . . 8  |-  ( ( a (,) b )  =/=  (/)  ->  ( F `  ( a (,) b
) )  =  <. a ,  b >. )
76fveq2d 5732 . . . . . . 7  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( (,) `  <. a ,  b >. )
)
8 df-ov 6084 . . . . . . 7  |-  ( a (,) b )  =  ( (,) `  <. a ,  b >. )
97, 8syl6eqr 2486 . . . . . 6  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( a (,) b ) )
10 df-ne 2601 . . . . . . . 8  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
11 neeq1 2609 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( A  =/=  (/)  <->  ( a (,) b )  =/=  (/) ) )
1210, 11syl5bbr 251 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  <->  ( a (,) b )  =/=  (/) ) )
13 fveq2 5728 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
1413fveq2d 5732 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( (,) `  ( F `  A ) )  =  ( (,) `  ( F `  ( a (,) b ) ) ) )
15 id 20 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  A  =  ( a (,) b ) )
1614, 15eqeq12d 2450 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  (
( (,) `  ( F `  A )
)  =  A  <->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( a (,) b ) ) )
1712, 16imbi12d 312 . . . . . 6  |-  ( A  =  ( a (,) b )  ->  (
( -.  A  =  (/)  ->  ( (,) `  ( F `  A )
)  =  A )  <-> 
( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  ( a (,) b
) ) )  =  ( a (,) b
) ) ) )
189, 17mpbiri 225 . . . . 5  |-  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A ) )  =  A ) )
1918a1i 11 . . . 4  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A ) )  =  A ) ) )
2019rexlimivv 2835 . . 3  |-  ( E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  A ) )
214, 20sylbi 188 . 2  |-  ( A  e.  ran  (,)  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A )
)  =  A ) )
22 ioorebas 11006 . . . . . . 7  |-  ( 0 (,) 0 )  e. 
ran  (,)
235ioorval 19466 . . . . . . 7  |-  ( ( 0 (,) 0 )  e.  ran  (,)  ->  ( F `  ( 0 (,) 0 ) )  =  if ( ( 0 (,) 0 )  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( ( 0 (,) 0 ) ,  RR* ,  `'  <  ) ,  sup ( ( 0 (,) 0 ) ,  RR* ,  <  ) >. )
)
2422, 23ax-mp 8 . . . . . 6  |-  ( F `
 ( 0 (,) 0 ) )  =  if ( ( 0 (,) 0 )  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
( 0 (,) 0
) ,  RR* ,  `'  <  ) ,  sup (
( 0 (,) 0
) ,  RR* ,  <  )
>. )
25 iooid 10944 . . . . . . 7  |-  ( 0 (,) 0 )  =  (/)
26 iftrue 3745 . . . . . . 7  |-  ( ( 0 (,) 0 )  =  (/)  ->  if ( ( 0 (,) 0
)  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( ( 0 (,) 0 ) , 
RR* ,  `'  <  ) ,  sup ( ( 0 (,) 0 ) ,  RR* ,  <  ) >. )  =  <. 0 ,  0 >. )
2725, 26ax-mp 8 . . . . . 6  |-  if ( ( 0 (,) 0
)  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( ( 0 (,) 0 ) , 
RR* ,  `'  <  ) ,  sup ( ( 0 (,) 0 ) ,  RR* ,  <  ) >. )  =  <. 0 ,  0 >.
2824, 27eqtri 2456 . . . . 5  |-  ( F `
 ( 0 (,) 0 ) )  = 
<. 0 ,  0
>.
2928fveq2i 5731 . . . 4  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( (,) `  <. 0 ,  0 >. )
30 df-ov 6084 . . . 4  |-  ( 0 (,) 0 )  =  ( (,) `  <. 0 ,  0 >. )
3129, 30eqtr4i 2459 . . 3  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( 0 (,) 0 )
3225eqeq2i 2446 . . . . . 6  |-  ( A  =  ( 0 (,) 0 )  <->  A  =  (/) )
3332biimpri 198 . . . . 5  |-  ( A  =  (/)  ->  A  =  ( 0 (,) 0
) )
3433fveq2d 5732 . . . 4  |-  ( A  =  (/)  ->  ( F `
 A )  =  ( F `  (
0 (,) 0 ) ) )
3534fveq2d 5732 . . 3  |-  ( A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  ( (,) `  ( F `
 ( 0 (,) 0 ) ) ) )
3631, 35, 333eqtr4a 2494 . 2  |-  ( A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  A )
3721, 36pm2.61d2 154 1  |-  ( A  e.  ran  (,)  ->  ( (,) `  ( F `
 A ) )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   (/)c0 3628   ifcif 3739   ~Pcpw 3799   <.cop 3817    e. cmpt 4266    X. cxp 4876   `'ccnv 4877   ran crn 4879    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   supcsup 7445   RRcr 8989   0cc0 8990   RR*cxr 9119    < clt 9120   (,)cioo 10916
This theorem is referenced by:  uniioombllem2  19475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-ioo 10920
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