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Theorem ovolficc 19357
Description: Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolficc  |-  ( ( 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 ovolficc
StepHypRef Expression
1 iccf 10995 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
2 inss2 3554 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 ressxr 9121 . . . . . . . . 9  |-  RR  C_  RR*
4 xpss12 4973 . . . . . . . . 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 3349 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
7 fss 5591 . . . . . . 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 5592 . . . . . 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 5583 . . . . 5  |-  ( ( [,]  o.  F ) : NN --> ~P RR*  ->  ( [,]  o.  F
)  Fn  NN )
12 fniunfv 5986 . . . . 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 3368 . . 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 3330 . . 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 3335 . . . . . 6  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
18 eliun 4089 . . . . . . 7  |-  ( z  e.  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n )  <->  E. n  e.  NN  z  e.  ( ( [,]  o.  F
) `  n )
)
19 fvco3 5792 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( [,]  o.  F
) `  n )  =  ( [,] `  ( F `  n )
) )
20 ffvelrn 5860 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
212, 20sseldi 3338 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
22 1st2nd2 6378 . . . . . . . . . . . . . . . 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 5724 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( [,] `  ( F `  n ) )  =  ( [,] `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
25 df-ov 6076 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) [,] ( 2nd `  ( F `  n )
) )  =  ( [,] `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
2624, 25syl6eqr 2485 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( [,] `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) [,] ( 2nd `  ( F `  n
) ) ) )
2719, 26eqtrd 2467 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( [,]  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) [,] ( 2nd `  ( F `  n ) ) ) )
2827eleq2d 2502 . . . . . . . . . . 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 19355 . . . . . . . . . . . 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 elicc2 10967 . . . . . . . . . . . . . 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
) ) ) ) )
31 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 )
) ) ) )
3230, 31syl6bb 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 )
) ) ) ) )
33323adant3 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 )
) ) ) ) )
3429, 33syl 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 )
) ) ) ) )
3528, 34bitrd 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 )
) ) ) ) )
3635adantll 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 )
) ) ) ) )
37 simpll 731 . . . . . . . . . 10  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  z  e.  RR )
3837biantrurd 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 )
) ) ) ) )
3936, 38bitr4d 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 )
) ) ) )
4039rexbidva 2714 . . . . . . 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 )
) ) ) )
4118, 40syl5bb 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 )
) ) ) )
4217, 41sylan 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 )
) ) ) )
4342an32s 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 )
) ) ) )
4443ralbidva 2713 . . 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 )
) ) ) )
4516, 44syl5bb 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
) ) ) ) )
4615, 45bitr3d 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   <.cop 3809   U.cuni 4007   U_ciun 4085   class class class wbr 4204    X. cxp 4868   ran crn 4871    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   RRcr 8981   RR*cxr 9111    <_ cle 9113   NNcn 9992   [,]cicc 10911
This theorem is referenced by:  ovollb2lem  19376  ovolctb  19378  ovolicc1  19404  ioombl1lem4  19447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-icc 10915
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