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Theorem ovolfcl 18826
Description: Closure for the interval endpoint function. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
ovolfcl  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( 1st `  ( F `  N )
)  e.  RR  /\  ( 2nd `  ( F `
 N ) )  e.  RR  /\  ( 1st `  ( F `  N ) )  <_ 
( 2nd `  ( F `  N )
) ) )

Proof of Theorem ovolfcl
StepHypRef Expression
1 inss2 3390 . . . . 5  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
2 ffvelrn 5663 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
31, 2sseldi 3178 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  ( RR  X.  RR ) )
4 1st2nd2 6159 . . . 4  |-  ( ( F `  N )  e.  ( RR  X.  RR )  ->  ( F `
 N )  = 
<. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >. )
53, 4syl 15 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  =  <. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >. )
65, 2eqeltrrd 2358 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  <. ( 1st `  ( F `  N ) ) ,  ( 2nd `  ( F `  N )
) >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
7 ancom 437 . . 3  |-  ( ( ( 1st `  ( F `  N )
)  <_  ( 2nd `  ( F `  N
) )  /\  (
( 1st `  ( F `  N )
)  e.  RR  /\  ( 2nd `  ( F `
 N ) )  e.  RR ) )  <-> 
( ( ( 1st `  ( F `  N
) )  e.  RR  /\  ( 2nd `  ( F `  N )
)  e.  RR )  /\  ( 1st `  ( F `  N )
)  <_  ( 2nd `  ( F `  N
) ) ) )
8 elin 3358 . . . 4  |-  ( <.
( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <->  ( <. ( 1st `  ( F `
 N ) ) ,  ( 2nd `  ( F `  N )
) >.  e.  <_  /\  <. ( 1st `  ( F `
 N ) ) ,  ( 2nd `  ( F `  N )
) >.  e.  ( RR 
X.  RR ) ) )
9 df-br 4024 . . . . . 6  |-  ( ( 1st `  ( F `
 N ) )  <_  ( 2nd `  ( F `  N )
)  <->  <. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  <_  )
109bicomi 193 . . . . 5  |-  ( <.
( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  <_  <->  ( 1st `  ( F `  N ) )  <_ 
( 2nd `  ( F `  N )
) )
11 opelxp 4719 . . . . 5  |-  ( <.
( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  ( RR  X.  RR )  <-> 
( ( 1st `  ( F `  N )
)  e.  RR  /\  ( 2nd `  ( F `
 N ) )  e.  RR ) )
1210, 11anbi12i 678 . . . 4  |-  ( (
<. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  <_  /\ 
<. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  ( RR  X.  RR ) )  <->  ( ( 1st `  ( F `  N
) )  <_  ( 2nd `  ( F `  N ) )  /\  ( ( 1st `  ( F `  N )
)  e.  RR  /\  ( 2nd `  ( F `
 N ) )  e.  RR ) ) )
138, 12bitri 240 . . 3  |-  ( <.
( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <->  ( ( 1st `  ( F `  N ) )  <_ 
( 2nd `  ( F `  N )
)  /\  ( ( 1st `  ( F `  N ) )  e.  RR  /\  ( 2nd `  ( F `  N
) )  e.  RR ) ) )
14 df-3an 936 . . 3  |-  ( ( ( 1st `  ( F `  N )
)  e.  RR  /\  ( 2nd `  ( F `
 N ) )  e.  RR  /\  ( 1st `  ( F `  N ) )  <_ 
( 2nd `  ( F `  N )
) )  <->  ( (
( 1st `  ( F `  N )
)  e.  RR  /\  ( 2nd `  ( F `
 N ) )  e.  RR )  /\  ( 1st `  ( F `
 N ) )  <_  ( 2nd `  ( F `  N )
) ) )
157, 13, 143bitr4i 268 . 2  |-  ( <.
( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <->  ( ( 1st `  ( F `  N ) )  e.  RR  /\  ( 2nd `  ( F `  N
) )  e.  RR  /\  ( 1st `  ( F `  N )
)  <_  ( 2nd `  ( F `  N
) ) ) )
166, 15sylib 188 1  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( 1st `  ( F `  N )
)  e.  RR  /\  ( 2nd `  ( F `
 N ) )  e.  RR  /\  ( 1st `  ( F `  N ) )  <_ 
( 2nd `  ( F `  N )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151   <.cop 3643   class class class wbr 4023    X. cxp 4687   -->wf 5251   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   RRcr 8736    <_ cle 8868   NNcn 9746
This theorem is referenced by:  ovolfioo  18827  ovolficc  18828  ovolfsval  18830  ovolfsf  18831  ovollb2lem  18847  ovolshftlem1  18868  ovolscalem1  18872  ioombl1lem1  18915  ioombl1lem3  18917  ioombl1lem4  18918  ovolfs2  18926  uniiccdif  18933  uniioovol  18934  uniioombllem2a  18937  uniioombllem2  18938  uniioombllem3a  18939  uniioombllem3  18940  uniioombllem4  18941  uniioombllem6  18943
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-1st 6122  df-2nd 6123
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