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Theorem ovolficcss 19233
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 5317 . . 3  |-  ran  ( [,]  o.  F )  =  ( [,] " ran  F )
2 inss2 3505 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 ffvelrn 5807 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
42, 3sseldi 3289 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  ( RR  X.  RR ) )
5 1st2nd2 6325 . . . . . . . . . . 11  |-  ( ( F `  y )  e.  ( RR  X.  RR )  ->  ( F `
 y )  = 
<. ( 1st `  ( F `  y )
) ,  ( 2nd `  ( F `  y
) ) >. )
64, 5syl 16 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  =  <. ( 1st `  ( F `  y )
) ,  ( 2nd `  ( F `  y
) ) >. )
76fveq2d 5672 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( [,] `  <. ( 1st `  ( F `
 y ) ) ,  ( 2nd `  ( F `  y )
) >. ) )
8 df-ov 6023 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 y ) ) [,] ( 2nd `  ( F `  y )
) )  =  ( [,] `  <. ( 1st `  ( F `  y ) ) ,  ( 2nd `  ( F `  y )
) >. )
97, 8syl6eqr 2437 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( ( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) ) )
10 xp1st 6315 . . . . . . . . . 10  |-  ( ( F `  y )  e.  ( RR  X.  RR )  ->  ( 1st `  ( F `  y
) )  e.  RR )
114, 10syl 16 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( 1st `  ( F `  y ) )  e.  RR )
12 xp2nd 6316 . . . . . . . . . 10  |-  ( ( F `  y )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( F `  y
) )  e.  RR )
134, 12syl 16 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( 2nd `  ( F `  y ) )  e.  RR )
14 iccssre 10924 . . . . . . . . 9  |-  ( ( ( 1st `  ( F `  y )
)  e.  RR  /\  ( 2nd `  ( F `
 y ) )  e.  RR )  -> 
( ( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) )  C_  RR )
1511, 13, 14syl2anc 643 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  (
( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) )  C_  RR )
169, 15eqsstrd 3325 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  C_  RR )
17 reex 9014 . . . . . . . 8  |-  RR  e.  _V
1817elpw2 4305 . . . . . . 7  |-  ( ( [,] `  ( F `
 y ) )  e.  ~P RR  <->  ( [,] `  ( F `  y
) )  C_  RR )
1916, 18sylibr 204 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  e. 
~P RR )
2019ralrimiva 2732 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. y  e.  NN  ( [,] `  ( F `  y )
)  e.  ~P RR )
21 ffn 5531 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
22 fveq2 5668 . . . . . . . 8  |-  ( x  =  ( F `  y )  ->  ( [,] `  x )  =  ( [,] `  ( F `  y )
) )
2322eleq1d 2453 . . . . . . 7  |-  ( x  =  ( F `  y )  ->  (
( [,] `  x
)  e.  ~P RR  <->  ( [,] `  ( F `
 y ) )  e.  ~P RR ) )
2423ralrn 5812 . . . . . 6  |-  ( F  Fn  NN  ->  ( A. x  e.  ran  F ( [,] `  x
)  e.  ~P RR  <->  A. y  e.  NN  ( [,] `  ( F `  y ) )  e. 
~P RR ) )
2521, 24syl 16 . . . . 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 224 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. x  e.  ran  F ( [,] `  x )  e.  ~P RR )
27 iccf 10935 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
28 ffun 5533 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
2927, 28ax-mp 8 . . . . 5  |-  Fun  [,]
30 frn 5537 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
31 ressxr 9062 . . . . . . . . 9  |-  RR  C_  RR*
32 xpss12 4921 . . . . . . . . 9  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
3331, 31, 32mp2an 654 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
342, 33sstri 3300 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
3527fdmi 5536 . . . . . . 7  |-  dom  [,]  =  ( RR*  X.  RR* )
3634, 35sseqtr4i 3324 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  dom  [,]
3730, 36syl6ss 3303 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  dom  [,] )
38 funimass4 5716 . . . . 5  |-  ( ( Fun  [,]  /\  ran  F  C_ 
dom  [,] )  ->  (
( [,] " ran  F )  C_  ~P RR  <->  A. x  e.  ran  F
( [,] `  x
)  e.  ~P RR ) )
3929, 37, 38sylancr 645 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( [,] " ran  F )  C_  ~P RR  <->  A. x  e.  ran  F
( [,] `  x
)  e.  ~P RR ) )
4026, 39mpbird 224 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( [,] " ran  F ) 
C_  ~P RR )
411, 40syl5eqss 3335 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  ( [,]  o.  F ) 
C_  ~P RR )
42 sspwuni 4117 . 2  |-  ( ran  ( [,]  o.  F
)  C_  ~P RR  <->  U.
ran  ( [,]  o.  F )  C_  RR )
4341, 42sylib 189 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   <.cop 3760   U.cuni 3957    X. cxp 4816   dom cdm 4818   ran crn 4819   "cima 4821    o. ccom 4822   Fun wfun 5388    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287   RRcr 8922   RR*cxr 9052    <_ cle 9054   NNcn 9932   [,]cicc 10851
This theorem is referenced by:  ovollb2lem  19251  ovollb2  19252  uniiccdif  19337  uniiccvol  19339  uniioombllem3  19344  uniioombllem4  19345  uniioombllem5  19346  uniiccmbl  19349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-pre-lttri 8997  ax-pre-lttrn 8998
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-icc 10855
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