MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ioorinv Structured version   Unicode version

Theorem ioorinv 21031
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 11379 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 ffn 5554 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3 ovelrn 6234 . . . 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 21030 . . . . . . . 8  |-  ( ( a (,) b )  =/=  (/)  ->  ( F `  ( a (,) b
) )  =  <. a ,  b >. )
76fveq2d 5690 . . . . . . 7  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( (,) `  <. a ,  b >. )
)
8 df-ov 6089 . . . . . . 7  |-  ( a (,) b )  =  ( (,) `  <. a ,  b >. )
97, 8syl6eqr 2488 . . . . . 6  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( a (,) b ) )
10 df-ne 2603 . . . . . . . 8  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
11 neeq1 2611 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( A  =/=  (/)  <->  ( a (,) b )  =/=  (/) ) )
1210, 11syl5bbr 259 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  <->  ( a (,) b )  =/=  (/) ) )
13 fveq2 5686 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
1413fveq2d 5690 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( (,) `  ( F `  A ) )  =  ( (,) `  ( F `  ( a (,) b ) ) ) )
15 id 22 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  A  =  ( a (,) b ) )
1614, 15eqeq12d 2452 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  (
( (,) `  ( F `  A )
)  =  A  <->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( a (,) b ) ) )
1712, 16imbi12d 320 . . . . . 6  |-  ( A  =  ( a (,) b )  ->  (
( -.  A  =  (/)  ->  ( (,) `  ( F `  A )
)  =  A )  <-> 
( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  ( a (,) b
) ) )  =  ( a (,) b
) ) ) )
189, 17mpbiri 233 . . . . 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 2841 . . 3  |-  ( E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  A ) )
214, 20sylbi 195 . 2  |-  ( A  e.  ran  (,)  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A )
)  =  A ) )
22 ioorebas 11383 . . . . . . 7  |-  ( 0 (,) 0 )  e. 
ran  (,)
235ioorval 21029 . . . . . . 7  |-  ( ( 0 (,) 0 )  e.  ran  (,)  ->  ( F `  ( 0 (,) 0 ) )  =  if ( ( 0 (,) 0 )  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( ( 0 (,) 0 ) ,  RR* ,  `'  <  ) ,  sup ( ( 0 (,) 0 ) ,  RR* ,  <  ) >. )
)
2422, 23ax-mp 5 . . . . . 6  |-  ( F `
 ( 0 (,) 0 ) )  =  if ( ( 0 (,) 0 )  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
( 0 (,) 0
) ,  RR* ,  `'  <  ) ,  sup (
( 0 (,) 0
) ,  RR* ,  <  )
>. )
25 iooid 11320 . . . . . . 7  |-  ( 0 (,) 0 )  =  (/)
2625iftruei 3793 . . . . . 6  |-  if ( ( 0 (,) 0
)  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( ( 0 (,) 0 ) , 
RR* ,  `'  <  ) ,  sup ( ( 0 (,) 0 ) ,  RR* ,  <  ) >. )  =  <. 0 ,  0 >.
2724, 26eqtri 2458 . . . . 5  |-  ( F `
 ( 0 (,) 0 ) )  = 
<. 0 ,  0
>.
2827fveq2i 5689 . . . 4  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( (,) `  <. 0 ,  0 >. )
29 df-ov 6089 . . . 4  |-  ( 0 (,) 0 )  =  ( (,) `  <. 0 ,  0 >. )
3028, 29eqtr4i 2461 . . 3  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( 0 (,) 0 )
3125eqeq2i 2448 . . . . . 6  |-  ( A  =  ( 0 (,) 0 )  <->  A  =  (/) )
3231biimpri 206 . . . . 5  |-  ( A  =  (/)  ->  A  =  ( 0 (,) 0
) )
3332fveq2d 5690 . . . 4  |-  ( A  =  (/)  ->  ( F `
 A )  =  ( F `  (
0 (,) 0 ) ) )
3433fveq2d 5690 . . 3  |-  ( A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  ( (,) `  ( F `
 ( 0 (,) 0 ) ) ) )
3530, 34, 323eqtr4a 2496 . 2  |-  ( A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  A )
3621, 35pm2.61d2 160 1  |-  ( A  e.  ran  (,)  ->  ( (,) `  ( F `
 A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711   (/)c0 3632   ifcif 3786   ~Pcpw 3855   <.cop 3878    e. cmpt 4345    X. cxp 4833   `'ccnv 4834   ran crn 4836    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   supcsup 7682   RRcr 9273   0cc0 9274   RR*cxr 9409    < clt 9410   (,)cioo 11292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-ioo 11296
This theorem is referenced by:  uniioombllem2  21038
  Copyright terms: Public domain W3C validator