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Theorem ovolfioo 18843
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 10757 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 inss2 3403 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 ressxr 8892 . . . . . . . . 9  |-  RR  C_  RR*
4 xpss12 4808 . . . . . . . . 9  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
53, 3, 4mp2an 653 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
62, 5sstri 3201 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
7 fss 5413 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
86, 7mpan2 652 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> ( RR*  X.  RR* ) )
9 fco 5414 . . . . . 6  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
101, 8, 9sylancr 644 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( (,)  o.  F ) : NN --> ~P RR )
11 ffn 5405 . . . . 5  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  ( (,)  o.  F
)  Fn  NN )
12 fniunfv 5789 . . . . 5  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  =  U. ran  ( (,)  o.  F
) )
1310, 11, 123syl 18 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  =  U. ran  ( (,)  o.  F
) )
1413sseq2d 3219 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( A  C_  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  A  C_  U. ran  ( (,)  o.  F ) ) )
1514adantl 452 . 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 3183 . . 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 3188 . . . . . 6  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
18 eliun 3925 . . . . . . 7  |-  ( z  e.  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  E. n  e.  NN  z  e.  ( ( (,)  o.  F
) `  n )
)
19 fvco3 5612 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
20 ffvelrn 5679 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
212, 20sseldi 3191 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
22 1st2nd2 6175 . . . . . . . . . . . . . . . 16  |-  ( ( F `  n )  e.  ( RR  X.  RR )  ->  ( F `
 n )  = 
<. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
2321, 22syl 15 . . . . . . . . . . . . . . 15  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
2423fveq2d 5545 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
25 df-ov 5877 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
2624, 25syl6eqr 2346 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
2719, 26eqtrd 2328 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
2827eleq2d 2363 . . . . . . . . . . 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 18842 . . . . . . . . . . . 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 8893 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) )  e.  RR  ->  ( 1st `  ( F `  n ) )  e. 
RR* )
31 rexr 8893 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( F `
 n ) )  e.  RR  ->  ( 2nd `  ( F `  n ) )  e. 
RR* )
32 elioo1 10712 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 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 938 . . . . . . . . . . . . . 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 252 . . . . . . . . . . . . 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 975 . . . . . . . . . . . 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 15 . . . . . . . . . . 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 244 . . . . . . . . . 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 694 . . . . . . . . 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 8893 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  RR* )
4140ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  z  e.  RR* )
4241biantrurd 494 . . . . . . . . 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 247 . . . . . . . 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 2573 . . . . . . 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 248 . . . . . 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 457 . . . . 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 779 . . . 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 2572 . . 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 248 . 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 246 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   <.cop 3656   U.cuni 3843   U_ciun 3921   class class class wbr 4039    X. cxp 4703   ran crn 4706    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   RRcr 8752   RR*cxr 8882    < clt 8883    <_ cle 8884   NNcn 9762   (,)cioo 10672
This theorem is referenced by:  ovollb2lem  18863  ovolunlem1  18872  ovoliunlem2  18878  ovolshftlem1  18884  ovolscalem1  18888  ioombl1lem4  18934
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-pre-lttri 8827  ax-pre-lttrn 8828
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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-ioo 10676
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