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Theorem ovolficc 21612
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 11619 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
2 inss2 3719 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 rexpssxrxp 9634 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
42, 3sstri 3513 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
5 fss 5737 . . . . . . 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 5739 . . . . . 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 5729 . . . . 5  |-  ( ( [,]  o.  F ) : NN --> ~P RR*  ->  ( [,]  o.  F
)  Fn  NN )
10 fniunfv 6145 . . . . 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 3532 . . 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 3494 . . 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 3499 . . . . . 6  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
16 eliun 4330 . . . . . . 7  |-  ( z  e.  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n )  <->  E. n  e.  NN  z  e.  ( ( [,]  o.  F
) `  n )
)
17 fvco3 5942 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( [,]  o.  F
) `  n )  =  ( [,] `  ( F `  n )
) )
18 ffvelrn 6017 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
192, 18sseldi 3502 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
20 1st2nd2 6818 . . . . . . . . . . . . . . . 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 5868 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( [,] `  ( F `  n ) )  =  ( [,] `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
23 df-ov 6285 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) [,] ( 2nd `  ( F `  n )
) )  =  ( [,] `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
2422, 23syl6eqr 2526 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( [,] `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) [,] ( 2nd `  ( F `  n
) ) ) )
2517, 24eqtrd 2508 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( [,]  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) [,] ( 2nd `  ( F `  n ) ) ) )
2625eleq2d 2537 . . . . . . . . . . 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 21610 . . . . . . . . . . . 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 elicc2 11585 . . . . . . . . . . . . . 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
) ) ) ) )
29 3anass 977 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR  /\  ( 1st `  ( F `
 n ) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) )  <->  ( z  e.  RR  /\  ( ( 1st `  ( F `
 n ) )  <_  z  /\  z  <_  ( 2nd `  ( F `  n )
) ) ) )
3028, 29syl6bb 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 )
) ) ) ) )
31303adant3 1016 . . . . . . . . . . . 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 )
) ) ) ) )
3227, 31syl 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 )
) ) ) ) )
3326, 32bitrd 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 )
) ) ) ) )
3433adantll 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 )
) ) ) ) )
35 simpll 753 . . . . . . . . . 10  |-  ( ( ( z  e.  RR  /\  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  /\  n  e.  NN )  ->  z  e.  RR )
3635biantrurd 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 )
) ) ) ) )
3734, 36bitr4d 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 )
) ) ) )
3837rexbidva 2970 . . . . . . 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 )
) ) ) )
3916, 38syl5bb 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 )
) ) ) )
4015, 39sylan 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 )
) ) ) )
4140an32s 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 )
) ) ) )
4241ralbidva 2900 . . 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 )
) ) ) )
4314, 42syl5bb 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
) ) ) ) )
4413, 43bitr3d 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   <.cop 4033   U.cuni 4245   U_ciun 4325   class class class wbr 4447    X. cxp 4997   ran crn 5000    o. ccom 5003    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   RRcr 9487   RR*cxr 9623    <_ cle 9625   NNcn 10532   [,]cicc 11528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-pre-lttri 9562  ax-pre-lttrn 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-icc 11532
This theorem is referenced by:  ovollb2lem  21631  ovolctb  21633  ovolicc1  21659  ioombl1lem4  21703
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