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Theorem ovolficcss 22173
Description: Any (closed) interval covering is a subset of the reals. (Contributed by Mario Carneiro, 24-Mar-2015.)
Assertion
Ref Expression
ovolficcss  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )

Proof of Theorem ovolficcss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnco2 5330 . . 3  |-  ran  ( [,]  o.  F )  =  ( [,] " ran  F )
2 inss2 3660 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 ffvelrn 6007 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
42, 3sseldi 3440 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  ( RR  X.  RR ) )
5 1st2nd2 6821 . . . . . . . . . . 11  |-  ( ( F `  y )  e.  ( RR  X.  RR )  ->  ( F `
 y )  = 
<. ( 1st `  ( F `  y )
) ,  ( 2nd `  ( F `  y
) ) >. )
64, 5syl 17 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  =  <. ( 1st `  ( F `  y )
) ,  ( 2nd `  ( F `  y
) ) >. )
76fveq2d 5853 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( [,] `  <. ( 1st `  ( F `
 y ) ) ,  ( 2nd `  ( F `  y )
) >. ) )
8 df-ov 6281 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 y ) ) [,] ( 2nd `  ( F `  y )
) )  =  ( [,] `  <. ( 1st `  ( F `  y ) ) ,  ( 2nd `  ( F `  y )
) >. )
97, 8syl6eqr 2461 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( ( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) ) )
10 xp1st 6814 . . . . . . . . . 10  |-  ( ( F `  y )  e.  ( RR  X.  RR )  ->  ( 1st `  ( F `  y
) )  e.  RR )
114, 10syl 17 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( 1st `  ( F `  y ) )  e.  RR )
12 xp2nd 6815 . . . . . . . . . 10  |-  ( ( F `  y )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( F `  y
) )  e.  RR )
134, 12syl 17 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( 2nd `  ( F `  y ) )  e.  RR )
14 iccssre 11660 . . . . . . . . 9  |-  ( ( ( 1st `  ( F `  y )
)  e.  RR  /\  ( 2nd `  ( F `
 y ) )  e.  RR )  -> 
( ( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) )  C_  RR )
1511, 13, 14syl2anc 659 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  (
( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) )  C_  RR )
169, 15eqsstrd 3476 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  C_  RR )
17 reex 9613 . . . . . . . 8  |-  RR  e.  _V
1817elpw2 4558 . . . . . . 7  |-  ( ( [,] `  ( F `
 y ) )  e.  ~P RR  <->  ( [,] `  ( F `  y
) )  C_  RR )
1916, 18sylibr 212 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  e. 
~P RR )
2019ralrimiva 2818 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. y  e.  NN  ( [,] `  ( F `  y )
)  e.  ~P RR )
21 ffn 5714 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
22 fveq2 5849 . . . . . . . 8  |-  ( x  =  ( F `  y )  ->  ( [,] `  x )  =  ( [,] `  ( F `  y )
) )
2322eleq1d 2471 . . . . . . 7  |-  ( x  =  ( F `  y )  ->  (
( [,] `  x
)  e.  ~P RR  <->  ( [,] `  ( F `
 y ) )  e.  ~P RR ) )
2423ralrn 6012 . . . . . 6  |-  ( F  Fn  NN  ->  ( A. x  e.  ran  F ( [,] `  x
)  e.  ~P RR  <->  A. y  e.  NN  ( [,] `  ( F `  y ) )  e. 
~P RR ) )
2521, 24syl 17 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( A. x  e.  ran  F ( [,] `  x
)  e.  ~P RR  <->  A. y  e.  NN  ( [,] `  ( F `  y ) )  e. 
~P RR ) )
2620, 25mpbird 232 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. x  e.  ran  F ( [,] `  x )  e.  ~P RR )
27 iccf 11677 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
28 ffun 5716 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
2927, 28ax-mp 5 . . . . 5  |-  Fun  [,]
30 frn 5720 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
31 rexpssxrxp 9668 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
322, 31sstri 3451 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
3327fdmi 5719 . . . . . . 7  |-  dom  [,]  =  ( RR*  X.  RR* )
3432, 33sseqtr4i 3475 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  dom  [,]
3530, 34syl6ss 3454 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  dom  [,] )
36 funimass4 5900 . . . . 5  |-  ( ( Fun  [,]  /\  ran  F  C_ 
dom  [,] )  ->  (
( [,] " ran  F )  C_  ~P RR  <->  A. x  e.  ran  F
( [,] `  x
)  e.  ~P RR ) )
3729, 35, 36sylancr 661 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( [,] " ran  F )  C_  ~P RR  <->  A. x  e.  ran  F
( [,] `  x
)  e.  ~P RR ) )
3826, 37mpbird 232 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( [,] " ran  F ) 
C_  ~P RR )
391, 38syl5eqss 3486 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  ( [,]  o.  F ) 
C_  ~P RR )
40 sspwuni 4360 . 2  |-  ( ran  ( [,]  o.  F
)  C_  ~P RR  <->  U.
ran  ( [,]  o.  F )  C_  RR )
4139, 40sylib 196 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754    i^i cin 3413    C_ wss 3414   ~Pcpw 3955   <.cop 3978   U.cuni 4191    X. cxp 4821   dom cdm 4823   ran crn 4824   "cima 4826    o. ccom 4827   Fun wfun 5563    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   RRcr 9521   RR*cxr 9657    <_ cle 9659   NNcn 10576   [,]cicc 11585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-pre-lttri 9596  ax-pre-lttrn 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-icc 11589
This theorem is referenced by:  ovollb2lem  22191  ovollb2  22192  uniiccdif  22279  uniiccvol  22281  uniioombllem3  22286  uniioombllem4  22287  uniioombllem5  22288  uniiccmbl  22291
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