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Theorem ovolficc 22006
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 11648 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
2 inss2 3715 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 rexpssxrxp 9655 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
42, 3sstri 3508 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
5 fss 5745 . . . . . . 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 5747 . . . . . 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 5737 . . . . 5  |-  ( ( [,]  o.  F ) : NN --> ~P RR*  ->  ( [,]  o.  F
)  Fn  NN )
10 fniunfv 6160 . . . . 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 3527 . . 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 3489 . . 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 3494 . . . . . 6  |-  ( ( A  C_  RR  /\  z  e.  A )  ->  z  e.  RR )
16 eliun 4337 . . . . . . 7  |-  ( z  e.  U_ n  e.  NN  ( ( [,] 
o.  F ) `  n )  <->  E. n  e.  NN  z  e.  ( ( [,]  o.  F
) `  n )
)
17 fvco3 5950 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( [,]  o.  F
) `  n )  =  ( [,] `  ( F `  n )
) )
18 ffvelrn 6030 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
192, 18sseldi 3497 . . . . . . . . . . . . . . . 16  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR  X.  RR ) )
20 1st2nd2 6836 . . . . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( [,] `  ( F `  n ) )  =  ( [,] `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
23 df-ov 6299 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( F `
 n ) ) [,] ( 2nd `  ( F `  n )
) )  =  ( [,] `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
2422, 23syl6eqr 2516 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( [,] `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) [,] ( 2nd `  ( F `  n
) ) ) )
2517, 24eqtrd 2498 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( [,]  o.  F
) `  n )  =  ( ( 1st `  ( F `  n
) ) [,] ( 2nd `  ( F `  n ) ) ) )
2625eleq2d 2527 . . . . . . . . . . 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 22004 . . . . . . . . . . . 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 11614 . . . . . . . . . . . . . 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 2965 . . . . . . 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 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 )
) ) ) )
4241ralbidva 2893 . . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   <.cop 4038   U.cuni 4251   U_ciun 4332   class class class wbr 4456    X. cxp 5006   ran crn 5009    o. ccom 5012    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   RRcr 9508   RR*cxr 9644    <_ cle 9646   NNcn 10556   [,]cicc 11557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-icc 11561
This theorem is referenced by:  ovollb2lem  22025  ovolctb  22027  ovolicc1  22053  ioombl1lem4  22097
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