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Theorem ioorinv 18947
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 10757 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 ffn 5405 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3 ovelrn 6012 . . . 4  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( A  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b ) ) )
41, 2, 3mp2b 9 . . 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 18946 . . . . . . . 8  |-  ( ( a (,) b )  =/=  (/)  ->  ( F `  ( a (,) b
) )  =  <. a ,  b >. )
76fveq2d 5545 . . . . . . 7  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( (,) `  <. a ,  b >. )
)
8 df-ov 5877 . . . . . . 7  |-  ( a (,) b )  =  ( (,) `  <. a ,  b >. )
97, 8syl6eqr 2346 . . . . . 6  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( a (,) b ) )
10 df-ne 2461 . . . . . . . 8  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
11 neeq1 2467 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( A  =/=  (/)  <->  ( a (,) b )  =/=  (/) ) )
1210, 11syl5bbr 250 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  <->  ( a (,) b )  =/=  (/) ) )
13 fveq2 5541 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
1413fveq2d 5545 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( (,) `  ( F `  A ) )  =  ( (,) `  ( F `  ( a (,) b ) ) ) )
15 id 19 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  A  =  ( a (,) b ) )
1614, 15eqeq12d 2310 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  (
( (,) `  ( F `  A )
)  =  A  <->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( a (,) b ) ) )
1712, 16imbi12d 311 . . . . . 6  |-  ( A  =  ( a (,) b )  ->  (
( -.  A  =  (/)  ->  ( (,) `  ( F `  A )
)  =  A )  <-> 
( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  ( a (,) b
) ) )  =  ( a (,) b
) ) ) )
189, 17mpbiri 224 . . . . 5  |-  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A ) )  =  A ) )
1918a1i 10 . . . 4  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A ) )  =  A ) ) )
2019rexlimivv 2685 . . 3  |-  ( E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  A ) )
214, 20sylbi 187 . 2  |-  ( A  e.  ran  (,)  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A )
)  =  A ) )
22 ioorebas 10761 . . . . . . 7  |-  ( 0 (,) 0 )  e. 
ran  (,)
235ioorval 18945 . . . . . . 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 10700 . . . . . . 7  |-  ( 0 (,) 0 )  =  (/)
26 iftrue 3584 . . . . . . 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 2316 . . . . 5  |-  ( F `
 ( 0 (,) 0 ) )  = 
<. 0 ,  0
>.
2928fveq2i 5544 . . . 4  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( (,) `  <. 0 ,  0 >. )
30 df-ov 5877 . . . 4  |-  ( 0 (,) 0 )  =  ( (,) `  <. 0 ,  0 >. )
3129, 30eqtr4i 2319 . . 3  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( 0 (,) 0 )
3225eqeq2i 2306 . . . . . 6  |-  ( A  =  ( 0 (,) 0 )  <->  A  =  (/) )
3332biimpri 197 . . . . 5  |-  ( A  =  (/)  ->  A  =  ( 0 (,) 0
) )
3433fveq2d 5545 . . . 4  |-  ( A  =  (/)  ->  ( F `
 A )  =  ( F `  (
0 (,) 0 ) ) )
3534fveq2d 5545 . . 3  |-  ( A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  ( (,) `  ( F `
 ( 0 (,) 0 ) ) ) )
3631, 35, 333eqtr4a 2354 . 2  |-  ( A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  A )
3721, 36pm2.61d2 152 1  |-  ( A  e.  ran  (,)  ->  ( (,) `  ( F `
 A ) )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   (/)c0 3468   ifcif 3578   ~Pcpw 3638   <.cop 3656    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753   RR*cxr 8882    < clt 8883   (,)cioo 10672
This theorem is referenced by:  uniioombllem2  18954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-ioo 10676
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