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Theorem ovolfcl 18879
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 3424 . . . . 5  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
2 ffvelrn 5701 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
31, 2sseldi 3212 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  ( RR  X.  RR ) )
4 1st2nd2 6201 . . . 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 2391 . 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 3392 . . . 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 4061 . . . . . 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 4756 . . . . 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 1633    e. wcel 1701    i^i cin 3185   <.cop 3677   class class class wbr 4060    X. cxp 4724   -->wf 5288   ` cfv 5292   1stc1st 6162   2ndc2nd 6163   RRcr 8781    <_ cle 8913   NNcn 9791
This theorem is referenced by:  ovolfioo  18880  ovolficc  18881  ovolfsval  18883  ovolfsf  18884  ovollb2lem  18900  ovolshftlem1  18921  ovolscalem1  18925  ioombl1lem1  18968  ioombl1lem3  18970  ioombl1lem4  18971  ovolfs2  18979  uniiccdif  18986  uniioovol  18987  uniioombllem2a  18990  uniioombllem2  18991  uniioombllem3a  18992  uniioombllem3  18993  uniioombllem4  18994  uniioombllem6  18996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-1st 6164  df-2nd 6165
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