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Theorem ovolficcss 19319
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 5336 . . 3  |-  ran  ( [,]  o.  F )  =  ( [,] " ran  F )
2 inss2 3522 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 ffvelrn 5827 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
42, 3sseldi 3306 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  ( RR  X.  RR ) )
5 1st2nd2 6345 . . . . . . . . . . 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 5691 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( [,] `  <. ( 1st `  ( F `
 y ) ) ,  ( 2nd `  ( F `  y )
) >. ) )
8 df-ov 6043 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 y ) ) [,] ( 2nd `  ( F `  y )
) )  =  ( [,] `  <. ( 1st `  ( F `  y ) ) ,  ( 2nd `  ( F `  y )
) >. )
97, 8syl6eqr 2454 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( ( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) ) )
10 xp1st 6335 . . . . . . . . . 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 6336 . . . . . . . . . 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 10948 . . . . . . . . 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 3342 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  C_  RR )
17 reex 9037 . . . . . . . 8  |-  RR  e.  _V
1817elpw2 4324 . . . . . . 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 2749 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. y  e.  NN  ( [,] `  ( F `  y )
)  e.  ~P RR )
21 ffn 5550 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
22 fveq2 5687 . . . . . . . 8  |-  ( x  =  ( F `  y )  ->  ( [,] `  x )  =  ( [,] `  ( F `  y )
) )
2322eleq1d 2470 . . . . . . 7  |-  ( x  =  ( F `  y )  ->  (
( [,] `  x
)  e.  ~P RR  <->  ( [,] `  ( F `
 y ) )  e.  ~P RR ) )
2423ralrn 5832 . . . . . 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 10959 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
28 ffun 5552 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
2927, 28ax-mp 8 . . . . 5  |-  Fun  [,]
30 frn 5556 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
31 ressxr 9085 . . . . . . . . 9  |-  RR  C_  RR*
32 xpss12 4940 . . . . . . . . 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 3317 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
3527fdmi 5555 . . . . . . 7  |-  dom  [,]  =  ( RR*  X.  RR* )
3634, 35sseqtr4i 3341 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  dom  [,]
3730, 36syl6ss 3320 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  dom  [,] )
38 funimass4 5736 . . . . 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 3352 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  ( [,]  o.  F ) 
C_  ~P RR )
42 sspwuni 4136 . 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 1721   A.wral 2666    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   <.cop 3777   U.cuni 3975    X. cxp 4835   dom cdm 4837   ran crn 4838   "cima 4840    o. ccom 4841   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   RRcr 8945   RR*cxr 9075    <_ cle 9077   NNcn 9956   [,]cicc 10875
This theorem is referenced by:  ovollb2lem  19337  ovollb2  19338  uniiccdif  19423  uniiccvol  19425  uniioombllem3  19430  uniioombllem4  19431  uniioombllem5  19432  uniiccmbl  19435
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-pre-lttri 9020  ax-pre-lttrn 9021
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-icc 10879
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