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Theorem ovolficcss 18829
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 5180 . . 3  |-  ran  ( [,]  o.  F )  =  ( [,] " ran  F )
2 inss2 3390 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
42, 3sseldi 3178 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  ( RR  X.  RR ) )
5 1st2nd2 6159 . . . . . . . . . . 11  |-  ( ( F `  y )  e.  ( RR  X.  RR )  ->  ( F `
 y )  = 
<. ( 1st `  ( F `  y )
) ,  ( 2nd `  ( F `  y
) ) >. )
64, 5syl 15 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  =  <. ( 1st `  ( F `  y )
) ,  ( 2nd `  ( F `  y
) ) >. )
76fveq2d 5529 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( [,] `  <. ( 1st `  ( F `
 y ) ) ,  ( 2nd `  ( F `  y )
) >. ) )
8 df-ov 5861 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 y ) ) [,] ( 2nd `  ( F `  y )
) )  =  ( [,] `  <. ( 1st `  ( F `  y ) ) ,  ( 2nd `  ( F `  y )
) >. )
97, 8syl6eqr 2333 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( ( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) ) )
10 xp1st 6149 . . . . . . . . . 10  |-  ( ( F `  y )  e.  ( RR  X.  RR )  ->  ( 1st `  ( F `  y
) )  e.  RR )
114, 10syl 15 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( 1st `  ( F `  y ) )  e.  RR )
12 xp2nd 6150 . . . . . . . . . 10  |-  ( ( F `  y )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( F `  y
) )  e.  RR )
134, 12syl 15 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( 2nd `  ( F `  y ) )  e.  RR )
14 iccssre 10731 . . . . . . . . 9  |-  ( ( ( 1st `  ( F `  y )
)  e.  RR  /\  ( 2nd `  ( F `
 y ) )  e.  RR )  -> 
( ( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) )  C_  RR )
1511, 13, 14syl2anc 642 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  (
( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) )  C_  RR )
169, 15eqsstrd 3212 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  C_  RR )
17 reex 8828 . . . . . . . 8  |-  RR  e.  _V
1817elpw2 4175 . . . . . . 7  |-  ( ( [,] `  ( F `
 y ) )  e.  ~P RR  <->  ( [,] `  ( F `  y
) )  C_  RR )
1916, 18sylibr 203 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  e. 
~P RR )
2019ralrimiva 2626 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. y  e.  NN  ( [,] `  ( F `  y )
)  e.  ~P RR )
21 ffn 5389 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
22 fveq2 5525 . . . . . . . 8  |-  ( x  =  ( F `  y )  ->  ( [,] `  x )  =  ( [,] `  ( F `  y )
) )
2322eleq1d 2349 . . . . . . 7  |-  ( x  =  ( F `  y )  ->  (
( [,] `  x
)  e.  ~P RR  <->  ( [,] `  ( F `
 y ) )  e.  ~P RR ) )
2423ralrn 5668 . . . . . 6  |-  ( F  Fn  NN  ->  ( A. x  e.  ran  F ( [,] `  x
)  e.  ~P RR  <->  A. y  e.  NN  ( [,] `  ( F `  y ) )  e. 
~P RR ) )
2521, 24syl 15 . . . . 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 223 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. x  e.  ran  F ( [,] `  x )  e.  ~P RR )
27 iccf 10742 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
28 ffun 5391 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
2927, 28ax-mp 8 . . . . 5  |-  Fun  [,]
30 frn 5395 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
31 ressxr 8876 . . . . . . . . 9  |-  RR  C_  RR*
32 xpss12 4792 . . . . . . . . 9  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
3331, 31, 32mp2an 653 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
342, 33sstri 3188 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
3527fdmi 5394 . . . . . . 7  |-  dom  [,]  =  ( RR*  X.  RR* )
3634, 35sseqtr4i 3211 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  dom  [,]
3730, 36syl6ss 3191 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  dom  [,] )
38 funimass4 5573 . . . . 5  |-  ( ( Fun  [,]  /\  ran  F  C_ 
dom  [,] )  ->  (
( [,] " ran  F )  C_  ~P RR  <->  A. x  e.  ran  F
( [,] `  x
)  e.  ~P RR ) )
3929, 37, 38sylancr 644 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( [,] " ran  F )  C_  ~P RR  <->  A. x  e.  ran  F
( [,] `  x
)  e.  ~P RR ) )
4026, 39mpbird 223 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( [,] " ran  F ) 
C_  ~P RR )
411, 40syl5eqss 3222 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  ( [,]  o.  F ) 
C_  ~P RR )
42 sspwuni 3987 . 2  |-  ( ran  ( [,]  o.  F
)  C_  ~P RR  <->  U.
ran  ( [,]  o.  F )  C_  RR )
4341, 42sylib 188 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   <.cop 3643   U.cuni 3827    X. cxp 4687   dom cdm 4689   ran crn 4690   "cima 4692    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RRcr 8736   RR*cxr 8866    <_ cle 8868   NNcn 9746   [,]cicc 10659
This theorem is referenced by:  ovollb2lem  18847  ovollb2  18848  uniiccdif  18933  uniiccvol  18935  uniioombllem3  18940  uniioombllem4  18941  uniioombllem5  18942  uniiccmbl  18945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663
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