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Theorem ovolfioo 19231
Description: Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolfioo  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A  C_  U. ran  ( (,)  o.  F )  <->  A. z  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) )
Distinct variable groups:    z, n, A    n, F, z

Proof of Theorem ovolfioo
StepHypRef Expression
1 ioof 10934 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 inss2 3505 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 ressxr 9062 . . . . . . . . 9  |-  RR  C_  RR*
4 xpss12 4921 . . . . . . . . 9  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
53, 3, 4mp2an 654 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
62, 5sstri 3300 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
7 fss 5539 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
86, 7mpan2 653 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> ( RR*  X.  RR* ) )
9 fco 5540 . . . . . 6  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
101, 8, 9sylancr 645 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( (,)  o.  F ) : NN --> ~P RR )
11 ffn 5531 . . . . 5  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  ( (,)  o.  F
)  Fn  NN )
12 fniunfv 5933 . . . . 5  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  =  U. ran  ( (,)  o.  F
) )
1310, 11, 123syl 19 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  =  U. ran  ( (,)  o.  F
) )
1413sseq2d 3319 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( A  C_  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  A  C_  U. ran  ( (,)  o.  F ) ) )
1514adantl 453 . 2  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A  C_  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  A  C_  U. ran  ( (,)  o.  F ) ) )
16 dfss3 3281 . . 3  |-  ( A 
C_  U_ n  e.  NN  ( ( (,)  o.  F ) `  n
)  <->  A. z  e.  A  z  e.  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n ) )
17 ssel2 3286 . . . . . 6  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
18 eliun 4039 . . . . . . 7  |-  ( z  e.  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  E. n  e.  NN  z  e.  ( ( (,)  o.  F
) `  n )
)
19 fvco3 5739 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
20 ffvelrn 5807 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
212, 20sseldi 3289 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
22 1st2nd2 6325 . . . . . . . . . . . . . . . 16  |-  ( ( F `  n )  e.  ( RR  X.  RR )  ->  ( F `
 n )  = 
<. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
2321, 22syl 16 . . . . . . . . . . . . . . 15  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
2423fveq2d 5672 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
25 df-ov 6023 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
2624, 25syl6eqr 2437 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
2719, 26eqtrd 2419 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
2827eleq2d 2454 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
z  e.  ( ( (,)  o.  F ) `
 n )  <->  z  e.  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) ) )
29 ovolfcl 19230 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR  /\  ( 1st `  ( F `  n ) )  <_ 
( 2nd `  ( F `  n )
) ) )
30 rexr 9063 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) )  e.  RR  ->  ( 1st `  ( F `  n ) )  e. 
RR* )
31 rexr 9063 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( F `
 n ) )  e.  RR  ->  ( 2nd `  ( F `  n ) )  e. 
RR* )
32 elioo1 10888 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  ( F `  n )
)  e.  RR*  /\  ( 2nd `  ( F `  n ) )  e. 
RR* )  ->  (
z  e.  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  <->  ( z  e.  RR*  /\  ( 1st `  ( F `  n
) )  <  z  /\  z  <  ( 2nd `  ( F `  n
) ) ) ) )
3330, 31, 32syl2an 464 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR )  -> 
( z  e.  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) )  <->  ( z  e.  RR*  /\  ( 1st `  ( F `  n
) )  <  z  /\  z  <  ( 2nd `  ( F `  n
) ) ) ) )
34 3anass 940 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR*  /\  ( 1st `  ( F `  n ) )  < 
z  /\  z  <  ( 2nd `  ( F `
 n ) ) )  <->  ( z  e. 
RR*  /\  ( ( 1st `  ( F `  n ) )  < 
z  /\  z  <  ( 2nd `  ( F `
 n ) ) ) ) )
3533, 34syl6bb 253 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR )  -> 
( z  e.  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) )  <->  ( z  e.  RR*  /\  ( ( 1st `  ( F `
 n ) )  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) ) )
36353adant3 977 . . . . . . . . . . . 12  |-  ( ( ( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR  /\  ( 1st `  ( F `  n ) )  <_ 
( 2nd `  ( F `  n )
) )  ->  (
z  e.  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  <->  ( z  e.  RR*  /\  ( ( 1st `  ( F `
 n ) )  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) ) )
3729, 36syl 16 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
z  e.  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  <->  ( z  e.  RR*  /\  ( ( 1st `  ( F `
 n ) )  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) ) )
3828, 37bitrd 245 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
z  e.  ( ( (,)  o.  F ) `
 n )  <->  ( z  e.  RR*  /\  ( ( 1st `  ( F `
 n ) )  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) ) )
3938adantll 695 . . . . . . . . 9  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  ( z  e.  ( ( (,)  o.  F
) `  n )  <->  ( z  e.  RR*  /\  (
( 1st `  ( F `  n )
)  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) ) )
40 rexr 9063 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  RR* )
4140ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  z  e.  RR* )
4241biantrurd 495 . . . . . . . . 9  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  ( ( ( 1st `  ( F `  n
) )  <  z  /\  z  <  ( 2nd `  ( F `  n
) ) )  <->  ( z  e.  RR*  /\  ( ( 1st `  ( F `
 n ) )  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) ) )
4339, 42bitr4d 248 . . . . . . . 8  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  ( z  e.  ( ( (,)  o.  F
) `  n )  <->  ( ( 1st `  ( F `  n )
)  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) )
4443rexbidva 2666 . . . . . . 7  |-  ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( E. n  e.  NN  z  e.  ( ( (,)  o.  F
) `  n )  <->  E. n  e.  NN  (
( 1st `  ( F `  n )
)  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) )
4518, 44syl5bb 249 . . . . . 6  |-  ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( z  e.  U_ n  e.  NN  (
( (,)  o.  F
) `  n )  <->  E. n  e.  NN  (
( 1st `  ( F `  n )
)  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) )
4617, 45sylan 458 . . . . 5  |-  ( ( ( A  C_  RR  /\  z  e.  A )  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  ->  ( z  e. 
U_ n  e.  NN  ( ( (,)  o.  F ) `  n
)  <->  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) )
4746an32s 780 . . . 4  |-  ( ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  z  e.  A )  ->  ( z  e.  U_ n  e.  NN  (
( (,)  o.  F
) `  n )  <->  E. n  e.  NN  (
( 1st `  ( F `  n )
)  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) )
4847ralbidva 2665 . . 3  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A. z  e.  A  z  e.  U_ n  e.  NN  (
( (,)  o.  F
) `  n )  <->  A. z  e.  A  E. n  e.  NN  (
( 1st `  ( F `  n )
)  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) )
4916, 48syl5bb 249 . 2  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A  C_  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  A. z  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n
) )  <  z  /\  z  <  ( 2nd `  ( F `  n
) ) ) ) )
5015, 49bitr3d 247 1  |-  ( ( A  C_  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( A  C_  U. ran  ( (,)  o.  F )  <->  A. z  e.  A  E. n  e.  NN  ( ( 1st `  ( F `  n )
)  <  z  /\  z  <  ( 2nd `  ( F `  n )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   <.cop 3760   U.cuni 3957   U_ciun 4035   class class class wbr 4153    X. cxp 4816   ran crn 4819    o. ccom 4822    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287   RRcr 8922   RR*cxr 9052    < clt 9053    <_ cle 9054   NNcn 9932   (,)cioo 10848
This theorem is referenced by:  ovollb2lem  19251  ovolunlem1  19260  ovoliunlem2  19266  ovolshftlem1  19272  ovolscalem1  19276  ioombl1lem4  19322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-pre-lttri 8997  ax-pre-lttrn 8998
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-ioo 10852
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