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Theorem ovolfioo 21856
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 11632 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 inss2 3704 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 rexpssxrxp 9641 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
42, 3sstri 3498 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
5 fss 5729 . . . . . . 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 5731 . . . . . 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 5721 . . . . 5  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  ( (,)  o.  F
)  Fn  NN )
10 fniunfv 6144 . . . . 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 3517 . . 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 3479 . . 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 3484 . . . . . 6  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
16 eliun 4320 . . . . . . 7  |-  ( z  e.  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  E. n  e.  NN  z  e.  ( ( (,)  o.  F
) `  n )
)
17 fvco3 5935 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
18 ffvelrn 6014 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
192, 18sseldi 3487 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
20 1st2nd2 6822 . . . . . . . . . . . . . . . 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 5860 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
23 df-ov 6284 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
2422, 23syl6eqr 2502 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
2517, 24eqtrd 2484 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
2625eleq2d 2513 . . . . . . . . . . 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 21855 . . . . . . . . . . . 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 9642 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) )  e.  RR  ->  ( 1st `  ( F `  n ) )  e. 
RR* )
29 rexr 9642 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( F `
 n ) )  e.  RR  ->  ( 2nd `  ( F `  n ) )  e. 
RR* )
30 elioo1 11579 . . . . . . . . . . . . . . 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 978 . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . 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 9642 . . . . . . . . . . 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 2951 . . . . . . 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 804 . . . 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 2879 . . 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 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    i^i cin 3460    C_ wss 3461   ~Pcpw 3997   <.cop 4020   U.cuni 4234   U_ciun 4315   class class class wbr 4437    X. cxp 4987   ran crn 4990    o. ccom 4993    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   RRcr 9494   RR*cxr 9630    < clt 9631    <_ cle 9632   NNcn 10543   (,)cioo 11539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-pre-lttri 9569  ax-pre-lttrn 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-ioo 11543
This theorem is referenced by:  ovollb2lem  21876  ovolunlem1  21885  ovoliunlem2  21891  ovolshftlem1  21897  ovolscalem1  21901  ioombl1lem4  21948
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