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Theorem ovolficcss 21084
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 5452 . . 3  |-  ran  ( [,]  o.  F )  =  ( [,] " ran  F )
2 inss2 3678 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
3 ffvelrn 5949 . . . . . . . . . . . 12  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
42, 3sseldi 3461 . . . . . . . . . . 11  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( F `  y )  e.  ( RR  X.  RR ) )
5 1st2nd2 6722 . . . . . . . . . . 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 5802 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( [,] `  <. ( 1st `  ( F `
 y ) ) ,  ( 2nd `  ( F `  y )
) >. ) )
8 df-ov 6202 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 y ) ) [,] ( 2nd `  ( F `  y )
) )  =  ( [,] `  <. ( 1st `  ( F `  y ) ) ,  ( 2nd `  ( F `  y )
) >. )
97, 8syl6eqr 2513 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  =  ( ( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) ) )
10 xp1st 6715 . . . . . . . . . 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 6716 . . . . . . . . . 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 11487 . . . . . . . . 9  |-  ( ( ( 1st `  ( F `  y )
)  e.  RR  /\  ( 2nd `  ( F `
 y ) )  e.  RR )  -> 
( ( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) )  C_  RR )
1511, 13, 14syl2anc 661 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  (
( 1st `  ( F `  y )
) [,] ( 2nd `  ( F `  y
) ) )  C_  RR )
169, 15eqsstrd 3497 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  y  e.  NN )  ->  ( [,] `  ( F `  y ) )  C_  RR )
17 reex 9483 . . . . . . . 8  |-  RR  e.  _V
1817elpw2 4563 . . . . . . 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 2829 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. y  e.  NN  ( [,] `  ( F `  y )
)  e.  ~P RR )
21 ffn 5666 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
22 fveq2 5798 . . . . . . . 8  |-  ( x  =  ( F `  y )  ->  ( [,] `  x )  =  ( [,] `  ( F `  y )
) )
2322eleq1d 2523 . . . . . . 7  |-  ( x  =  ( F `  y )  ->  (
( [,] `  x
)  e.  ~P RR  <->  ( [,] `  ( F `
 y ) )  e.  ~P RR ) )
2423ralrn 5954 . . . . . 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 232 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  A. x  e.  ran  F ( [,] `  x )  e.  ~P RR )
27 iccf 11504 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
28 ffun 5668 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
2927, 28ax-mp 5 . . . . 5  |-  Fun  [,]
30 frn 5672 . . . . . 6  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
31 rexpssxrxp 9538 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
322, 31sstri 3472 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
3327fdmi 5671 . . . . . . 7  |-  dom  [,]  =  ( RR*  X.  RR* )
3432, 33sseqtr4i 3496 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  dom  [,]
3530, 34syl6ss 3475 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  dom  [,] )
36 funimass4 5850 . . . . 5  |-  ( ( Fun  [,]  /\  ran  F  C_ 
dom  [,] )  ->  (
( [,] " ran  F )  C_  ~P RR  <->  A. x  e.  ran  F
( [,] `  x
)  e.  ~P RR ) )
3729, 35, 36sylancr 663 . . . 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 3507 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  ( [,]  o.  F ) 
C_  ~P RR )
40 sspwuni 4363 . 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 369    = wceq 1370    e. wcel 1758   A.wral 2798    i^i cin 3434    C_ wss 3435   ~Pcpw 3967   <.cop 3990   U.cuni 4198    X. cxp 4945   dom cdm 4947   ran crn 4948   "cima 4950    o. ccom 4951   Fun wfun 5519    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6199   1stc1st 6684   2ndc2nd 6685   RRcr 9391   RR*cxr 9527    <_ cle 9529   NNcn 10432   [,]cicc 11413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-pre-lttri 9466  ax-pre-lttrn 9467
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-icc 11417
This theorem is referenced by:  ovollb2lem  21102  ovollb2  21103  uniiccdif  21190  uniiccvol  21192  uniioombllem3  21197  uniioombllem4  21198  uniioombllem5  21199  uniiccmbl  21202
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