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Theorem ioorinv 22111
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 11647 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 ffn 5737 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3 ovelrn 6450 . . . 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 22110 . . . . . . . 8  |-  ( ( a (,) b )  =/=  (/)  ->  ( F `  ( a (,) b
) )  =  <. a ,  b >. )
76fveq2d 5876 . . . . . . 7  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( (,) `  <. a ,  b >. )
)
8 df-ov 6299 . . . . . . 7  |-  ( a (,) b )  =  ( (,) `  <. a ,  b >. )
97, 8syl6eqr 2516 . . . . . 6  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( a (,) b ) )
10 df-ne 2654 . . . . . . . 8  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
11 neeq1 2738 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( A  =/=  (/)  <->  ( a (,) b )  =/=  (/) ) )
1210, 11syl5bbr 259 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  <->  ( a (,) b )  =/=  (/) ) )
13 fveq2 5872 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
1413fveq2d 5876 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( (,) `  ( F `  A ) )  =  ( (,) `  ( F `  ( a (,) b ) ) ) )
15 id 22 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  A  =  ( a (,) b ) )
1614, 15eqeq12d 2479 . . . . . . 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 2954 . . 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 11651 . . . . . . 7  |-  ( 0 (,) 0 )  e. 
ran  (,)
235ioorval 22109 . . . . . . 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 11582 . . . . . . 7  |-  ( 0 (,) 0 )  =  (/)
2625iftruei 3951 . . . . . 6  |-  if ( ( 0 (,) 0
)  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( ( 0 (,) 0 ) , 
RR* ,  `'  <  ) ,  sup ( ( 0 (,) 0 ) ,  RR* ,  <  ) >. )  =  <. 0 ,  0 >.
2724, 26eqtri 2486 . . . . 5  |-  ( F `
 ( 0 (,) 0 ) )  = 
<. 0 ,  0
>.
2827fveq2i 5875 . . . 4  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( (,) `  <. 0 ,  0 >. )
29 df-ov 6299 . . . 4  |-  ( 0 (,) 0 )  =  ( (,) `  <. 0 ,  0 >. )
3028, 29eqtr4i 2489 . . 3  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( 0 (,) 0 )
3125eqeq2i 2475 . . . . . 6  |-  ( A  =  ( 0 (,) 0 )  <->  A  =  (/) )
3231biimpri 206 . . . . 5  |-  ( A  =  (/)  ->  A  =  ( 0 (,) 0
) )
3332fveq2d 5876 . . . 4  |-  ( A  =  (/)  ->  ( F `
 A )  =  ( F `  (
0 (,) 0 ) ) )
3433fveq2d 5876 . . 3  |-  ( A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  ( (,) `  ( F `
 ( 0 (,) 0 ) ) ) )
3530, 34, 323eqtr4a 2524 . 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 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   (/)c0 3793   ifcif 3944   ~Pcpw 4015   <.cop 4038    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   ran crn 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   RRcr 9508   0cc0 9509   RR*cxr 9644    < clt 9645   (,)cioo 11554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-ioo 11558
This theorem is referenced by:  uniioombllem2  22118
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