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Theorem ovolfcl 21705
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 3719 . . . . 5  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
2 ffvelrn 6020 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
31, 2sseldi 3502 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  ( RR  X.  RR ) )
4 1st2nd2 6822 . . . 4  |-  ( ( F `  N )  e.  ( RR  X.  RR )  ->  ( F `
 N )  = 
<. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >. )
53, 4syl 16 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  =  <. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >. )
65, 2eqeltrrd 2556 . 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 450 . . 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 3687 . . . 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 4448 . . . . . 6  |-  ( ( 1st `  ( F `
 N ) )  <_  ( 2nd `  ( F `  N )
)  <->  <. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  <_  )
109bicomi 202 . . . . 5  |-  ( <.
( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  <_  <->  ( 1st `  ( F `  N ) )  <_ 
( 2nd `  ( F `  N )
) )
11 opelxp 5029 . . . . 5  |-  ( <.
( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >.  e.  ( RR  X.  RR )  <-> 
( ( 1st `  ( F `  N )
)  e.  RR  /\  ( 2nd `  ( F `
 N ) )  e.  RR ) )
1210, 11anbi12i 697 . . . 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 249 . . 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 975 . . 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 277 . 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 196 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475   <.cop 4033   class class class wbr 4447    X. cxp 4997   -->wf 5584   ` cfv 5588   1stc1st 6783   2ndc2nd 6784   RRcr 9492    <_ cle 9630   NNcn 10537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-1st 6785  df-2nd 6786
This theorem is referenced by:  ovolfioo  21706  ovolficc  21707  ovolfsval  21709  ovolfsf  21710  ovollb2lem  21726  ovolshftlem1  21747  ovolscalem1  21751  ioombl1lem1  21795  ioombl1lem3  21797  ioombl1lem4  21798  ovolfs2  21807  uniiccdif  21814  uniioovol  21815  uniioombllem2a  21818  uniioombllem2  21819  uniioombllem3a  21820  uniioombllem3  21821  uniioombllem4  21822  uniioombllem6  21824
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