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Theorem ovolfioo 21964
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 11543 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 inss2 3633 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 rexpssxrxp 9549 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
42, 3sstri 3426 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
5 fss 5647 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
64, 5mpan2 669 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> ( RR*  X.  RR* ) )
7 fco 5649 . . . . . 6  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
81, 6, 7sylancr 661 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( (,)  o.  F ) : NN --> ~P RR )
9 ffn 5639 . . . . 5  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  ( (,)  o.  F
)  Fn  NN )
10 fniunfv 6060 . . . . 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 3445 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( A  C_  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  A  C_  U. ran  ( (,)  o.  F ) ) )
1312adantl 464 . 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 3407 . . 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 3412 . . . . . 6  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
16 eliun 4248 . . . . . . 7  |-  ( z  e.  U_ n  e.  NN  ( ( (,) 
o.  F ) `  n )  <->  E. n  e.  NN  z  e.  ( ( (,)  o.  F
) `  n )
)
17 fvco3 5851 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( (,) `  ( F `  n )
) )
18 ffvelrn 5931 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
192, 18sseldi 3415 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
20 1st2nd2 6736 . . . . . . . . . . . . . . . 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 5778 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
23 df-ov 6199 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
2422, 23syl6eqr 2441 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
2517, 24eqtrd 2423 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( (,)  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) (,) ( 2nd `  ( F `  n ) ) ) )
2625eleq2d 2452 . . . . . . . . . . 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 21963 . . . . . . . . . . . 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 9550 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( F `
 n ) )  e.  RR  ->  ( 1st `  ( F `  n ) )  e. 
RR* )
29 rexr 9550 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( F `
 n ) )  e.  RR  ->  ( 2nd `  ( F `  n ) )  e. 
RR* )
30 elioo1 11490 . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . 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 975 . . . . . . . . . . . . . 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 1014 . . . . . . . . . . . 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 711 . . . . . . . . 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 9550 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  RR* )
3938ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  z  e.  RR* )
4039biantrurd 506 . . . . . . . . 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 2890 . . . . . . 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 469 . . . . 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 2818 . . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   E.wrex 2733    i^i cin 3388    C_ wss 3389   ~Pcpw 3927   <.cop 3950   U.cuni 4163   U_ciun 4243   class class class wbr 4367    X. cxp 4911   ran crn 4914    o. ccom 4917    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698   RRcr 9402   RR*cxr 9538    < clt 9539    <_ cle 9540   NNcn 10452   (,)cioo 11450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-pre-lttri 9477  ax-pre-lttrn 9478
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-ioo 11454
This theorem is referenced by:  ovollb2lem  21984  ovolunlem1  21993  ovoliunlem2  21999  ovolshftlem1  22005  ovolscalem1  22009  ioombl1lem4  22056
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