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Theorem ovolfioo 20926
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 11379 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 inss2 3566 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 rexpssxrxp 9420 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
42, 3sstri 3360 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
5 fss 5562 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
64, 5mpan2 671 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> ( RR*  X.  RR* ) )
7 fco 5563 . . . . . 6  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
81, 6, 7sylancr 663 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( (,)  o.  F ) : NN --> ~P RR )
9 ffn 5554 . . . . 5  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  ( (,)  o.  F
)  Fn  NN )
10 fniunfv 5959 . . . . 5  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  =  U. ran  ( (,)  o.  F
) )
118, 9, 103syl 20 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  =  U. ran  ( (,)  o.  F
) )
1211sseq2d 3379 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( A  C_  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  A  C_  U. ran  ( (,)  o.  F ) ) )
1312adantl 466 . 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 ) ) )
14 dfss3 3341 . . 3  |-  ( A 
C_  U_ n  e.  NN  ( ( (,)  o.  F ) `  n
)  <->  A. z  e.  A  z  e.  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n ) )
15 ssel2 3346 . . . . . 6  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
16 eliun 4170 . . . . . . 7  |-  ( z  e.  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  E. n  e.  NN  z  e.  ( ( (,)  o.  F
) `  n )
)
17 fvco3 5763 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
18 ffvelrn 5836 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
192, 18sseldi 3349 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
20 1st2nd2 6608 . . . . . . . . . . . . . . . 16  |-  ( ( F `  n )  e.  ( RR  X.  RR )  ->  ( F `
 n )  = 
<. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
2119, 20syl 16 . . . . . . . . . . . . . . 15  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
2221fveq2d 5690 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
23 df-ov 6089 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
2422, 23syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
2517, 24eqtrd 2470 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
2625eleq2d 2505 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
z  e.  ( ( (,)  o.  F ) `
 n )  <->  z  e.  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) ) )
27 ovolfcl 20925 . . . . . . . . . . . 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 )
) ) )
28 rexr 9421 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) )  e.  RR  ->  ( 1st `  ( F `  n ) )  e. 
RR* )
29 rexr 9421 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( F `
 n ) )  e.  RR  ->  ( 2nd `  ( F `  n ) )  e. 
RR* )
30 elioo1 11332 . . . . . . . . . . . . . . 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
) ) ) ) )
3128, 29, 30syl2an 477 . . . . . . . . . . . . . 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
) ) ) ) )
32 3anass 969 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR*  /\  ( 1st `  ( F `  n ) )  < 
z  /\  z  <  ( 2nd `  ( F `
 n ) ) )  <->  ( z  e. 
RR*  /\  ( ( 1st `  ( F `  n ) )  < 
z  /\  z  <  ( 2nd `  ( F `
 n ) ) ) ) )
3331, 32syl6bb 261 . . . . . . . . . . . . 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 )
) ) ) ) )
34333adant3 1008 . . . . . . . . . . . 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 )
) ) ) ) )
3527, 34syl 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 )
) ) ) ) )
3626, 35bitrd 253 . . . . . . . . . 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 )
) ) ) ) )
3736adantll 713 . . . . . . . . 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 )
) ) ) ) )
38 rexr 9421 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  RR* )
3938ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  z  e.  RR* )
4039biantrurd 508 . . . . . . . . 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 )
) ) ) ) )
4137, 40bitr4d 256 . . . . . . . 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 )
) ) ) )
4241rexbidva 2727 . . . . . . 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 )
) ) ) )
4316, 42syl5bb 257 . . . . . 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 )
) ) ) )
4415, 43sylan 471 . . . . 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 )
) ) ) )
4544an32s 802 . . . 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 )
) ) ) )
4645ralbidva 2726 . . 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 )
) ) ) )
4714, 46syl5bb 257 . 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
) ) ) ) )
4813, 47bitr3d 255 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   <.cop 3878   U.cuni 4086   U_ciun 4166   class class class wbr 4287    X. cxp 4833   ran crn 4836    o. ccom 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571   RRcr 9273   RR*cxr 9409    < clt 9410    <_ cle 9411   NNcn 10314   (,)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-pre-lttri 9348  ax-pre-lttrn 9349
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-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  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-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-ioo 11296
This theorem is referenced by:  ovollb2lem  20946  ovolunlem1  20955  ovoliunlem2  20961  ovolshftlem1  20967  ovolscalem1  20971  ioombl1lem4  21017
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