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Theorem ovolfcl 20972
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 3592 . . . . 5  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
2 ffvelrn 5862 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
31, 2sseldi 3375 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  ( RR  X.  RR ) )
4 1st2nd2 6634 . . . 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 2518 . 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 3560 . . . 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 4314 . . . . . 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 4890 . . . . 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 967 . . 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 965    = wceq 1369    e. wcel 1756    i^i cin 3348   <.cop 3904   class class class wbr 4313    X. cxp 4859   -->wf 5435   ` cfv 5439   1stc1st 6596   2ndc2nd 6597   RRcr 9302    <_ cle 9440   NNcn 10343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-1st 6598  df-2nd 6599
This theorem is referenced by:  ovolfioo  20973  ovolficc  20974  ovolfsval  20976  ovolfsf  20977  ovollb2lem  20993  ovolshftlem1  21014  ovolscalem1  21018  ioombl1lem1  21061  ioombl1lem3  21063  ioombl1lem4  21064  ovolfs2  21073  uniiccdif  21080  uniioovol  21081  uniioombllem2a  21084  uniioombllem2  21085  uniioombllem3a  21086  uniioombllem3  21087  uniioombllem4  21088  uniioombllem6  21090
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