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Theorem uniiccdif 19472
Description: A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)
Hypothesis
Ref Expression
uniioombl.1  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
Assertion
Ref Expression
uniiccdif  |-  ( ph  ->  ( U. ran  ( (,)  o.  F )  C_  U.
ran  ( [,]  o.  F )  /\  ( vol * `  ( U. ran  ( [,]  o.  F
)  \  U. ran  ( (,)  o.  F ) ) )  =  0 ) )

Proof of Theorem uniiccdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssun1 3512 . . 3  |-  U. ran  ( (,)  o.  F ) 
C_  ( U. ran  ( (,)  o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
2 uniioombl.1 . . . . . . . 8  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3 ovolfcl 19365 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 1st `  ( F `  x )
)  e.  RR  /\  ( 2nd `  ( F `
 x ) )  e.  RR  /\  ( 1st `  ( F `  x ) )  <_ 
( 2nd `  ( F `  x )
) ) )
42, 3sylan 459 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st `  ( F `
 x ) )  e.  RR  /\  ( 2nd `  ( F `  x ) )  e.  RR  /\  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) ) )
5 rexr 9132 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) )  e.  RR  ->  ( 1st `  ( F `  x ) )  e. 
RR* )
6 rexr 9132 . . . . . . . 8  |-  ( ( 2nd `  ( F `
 x ) )  e.  RR  ->  ( 2nd `  ( F `  x ) )  e. 
RR* )
7 id 21 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) )  <_  ( 2nd `  ( F `  x )
)  ->  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) )
8 prunioo 11027 . . . . . . . 8  |-  ( ( ( 1st `  ( F `  x )
)  e.  RR*  /\  ( 2nd `  ( F `  x ) )  e. 
RR*  /\  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) )  ->  ( ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  u.  {
( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )  =  ( ( 1st `  ( F `  x
) ) [,] ( 2nd `  ( F `  x ) ) ) )
95, 6, 7, 8syl3an 1227 . . . . . . 7  |-  ( ( ( 1st `  ( F `  x )
)  e.  RR  /\  ( 2nd `  ( F `
 x ) )  e.  RR  /\  ( 1st `  ( F `  x ) )  <_ 
( 2nd `  ( F `  x )
) )  ->  (
( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) )  u. 
{ ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )  =  ( ( 1st `  ( F `  x
) ) [,] ( 2nd `  ( F `  x ) ) ) )
104, 9syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) )  u. 
{ ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )  =  ( ( 1st `  ( F `  x
) ) [,] ( 2nd `  ( F `  x ) ) ) )
11 fvco3 5802 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
122, 11sylan 459 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( (,) `  ( F `  x )
) )
13 inss2 3564 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
142ffvelrnda 5872 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1513, 14sseldi 3348 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  ( RR  X.  RR ) )
16 1st2nd2 6388 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( RR  X.  RR )  ->  ( F `
 x )  = 
<. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1715, 16syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  = 
<. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1817fveq2d 5734 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
19 df-ov 6086 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
2018, 19syl6eqr 2488 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
2112, 20eqtrd 2470 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
22 df-pr 3823 . . . . . . . 8  |-  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )
23 fvco3 5802 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 1st  o.  F
) `  x )  =  ( 1st `  ( F `  x )
) )
242, 23sylan 459 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st  o.  F ) `
 x )  =  ( 1st `  ( F `  x )
) )
25 fvco3 5802 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 2nd  o.  F
) `  x )  =  ( 2nd `  ( F `  x )
) )
262, 25sylan 459 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd  o.  F ) `
 x )  =  ( 2nd `  ( F `  x )
) )
2724, 26preq12d 3893 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
2822, 27syl5eqr 2484 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( { ( ( 1st  o.  F ) `  x
) }  u.  {
( ( 2nd  o.  F ) `  x
) } )  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
2921, 28uneq12d 3504 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( (,)  o.  F
) `  x )  u.  ( { ( ( 1st  o.  F ) `
 x ) }  u.  { ( ( 2nd  o.  F ) `
 x ) } ) )  =  ( ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) )  u. 
{ ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } ) )
30 fvco3 5802 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
312, 30sylan 459 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( [,] `  ( F `  x )
) )
3217fveq2d 5734 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
33 df-ov 6086 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
3432, 33syl6eqr 2488 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3531, 34eqtrd 2470 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3610, 29, 353eqtr4rd 2481 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( ( (,) 
o.  F ) `  x )  u.  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } ) ) )
3736iuneq2dv 4116 . . . 4  |-  ( ph  ->  U_ x  e.  NN  ( ( [,]  o.  F ) `  x
)  =  U_ x  e.  NN  ( ( ( (,)  o.  F ) `
 x )  u.  ( { ( ( 1st  o.  F ) `
 x ) }  u.  { ( ( 2nd  o.  F ) `
 x ) } ) ) )
38 iccf 11005 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
39 ffn 5593 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4038, 39ax-mp 8 . . . . . 6  |-  [,]  Fn  ( RR*  X.  RR* )
41 ressxr 9131 . . . . . . . . 9  |-  RR  C_  RR*
42 xpss12 4983 . . . . . . . . 9  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
4341, 41, 42mp2an 655 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
4413, 43sstri 3359 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
45 fss 5601 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
462, 44, 45sylancl 645 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
47 fnfco 5611 . . . . . 6  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( [,]  o.  F )  Fn  NN )
4840, 46, 47sylancr 646 . . . . 5  |-  ( ph  ->  ( [,]  o.  F
)  Fn  NN )
49 fniunfv 5996 . . . . 5  |-  ( ( [,]  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x )  =  U. ran  ( [,]  o.  F
) )
5048, 49syl 16 . . . 4  |-  ( ph  ->  U_ x  e.  NN  ( ( [,]  o.  F ) `  x
)  =  U. ran  ( [,]  o.  F ) )
51 iunun 4173 . . . . 5  |-  U_ x  e.  NN  ( ( ( (,)  o.  F ) `
 x )  u.  ( { ( ( 1st  o.  F ) `
 x ) }  u.  { ( ( 2nd  o.  F ) `
 x ) } ) )  =  (
U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  u.  U_ x  e.  NN  ( { ( ( 1st  o.  F
) `  x ) }  u.  { (
( 2nd  o.  F
) `  x ) } ) )
52 ioof 11004 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
53 ffn 5593 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
5452, 53ax-mp 8 . . . . . . . 8  |-  (,)  Fn  ( RR*  X.  RR* )
55 fnfco 5611 . . . . . . . 8  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( (,)  o.  F )  Fn  NN )
5654, 46, 55sylancr 646 . . . . . . 7  |-  ( ph  ->  ( (,)  o.  F
)  Fn  NN )
57 fniunfv 5996 . . . . . . 7  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( (,) 
o.  F ) `  x )  =  U. ran  ( (,)  o.  F
) )
5856, 57syl 16 . . . . . 6  |-  ( ph  ->  U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  =  U. ran  ( (,)  o.  F ) )
59 iunun 4173 . . . . . . 7  |-  U_ x  e.  NN  ( { ( ( 1st  o.  F
) `  x ) }  u.  { (
( 2nd  o.  F
) `  x ) } )  =  (
U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  u.  U_ x  e.  NN  { ( ( 2nd  o.  F
) `  x ) } )
60 fo1st 6368 . . . . . . . . . . . . . 14  |-  1st : _V -onto-> _V
61 fofn 5657 . . . . . . . . . . . . . 14  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
6260, 61ax-mp 8 . . . . . . . . . . . . 13  |-  1st  Fn  _V
63 ssv 3370 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  _V
64 fss 5601 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  _V )  ->  F : NN --> _V )
652, 63, 64sylancl 645 . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> _V )
66 fnfco 5611 . . . . . . . . . . . . 13  |-  ( ( 1st  Fn  _V  /\  F : NN --> _V )  ->  ( 1st  o.  F
)  Fn  NN )
6762, 65, 66sylancr 646 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st  o.  F
)  Fn  NN )
68 fnfun 5544 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  Fun  ( 1st  o.  F ) )
6967, 68syl 16 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 1st  o.  F ) )
70 fndm 5546 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  dom  ( 1st  o.  F )  =  NN )
71 eqimss2 3403 . . . . . . . . . . . 12  |-  ( dom  ( 1st  o.  F
)  =  NN  ->  NN  C_  dom  ( 1st  o.  F ) )
7267, 70, 713syl 19 . . . . . . . . . . 11  |-  ( ph  ->  NN  C_  dom  ( 1st 
o.  F ) )
73 dfimafn2 5778 . . . . . . . . . . 11  |-  ( ( Fun  ( 1st  o.  F )  /\  NN  C_ 
dom  ( 1st  o.  F ) )  -> 
( ( 1st  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 1st  o.  F ) `
 x ) } )
7469, 72, 73syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 1st  o.  F ) `
 x ) } )
75 fnima 5565 . . . . . . . . . . 11  |-  ( ( 1st  o.  F )  Fn  NN  ->  (
( 1st  o.  F
) " NN )  =  ran  ( 1st 
o.  F ) )
7667, 75syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  o.  F ) " NN )  =  ran  ( 1st 
o.  F ) )
7774, 76eqtr3d 2472 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ran  ( 1st  o.  F ) )
78 rnco2 5379 . . . . . . . . 9  |-  ran  ( 1st  o.  F )  =  ( 1st " ran  F )
7977, 78syl6eq 2486 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ( 1st " ran  F
) )
80 fo2nd 6369 . . . . . . . . . . . . . 14  |-  2nd : _V -onto-> _V
81 fofn 5657 . . . . . . . . . . . . . 14  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
8280, 81ax-mp 8 . . . . . . . . . . . . 13  |-  2nd  Fn  _V
83 fnfco 5611 . . . . . . . . . . . . 13  |-  ( ( 2nd  Fn  _V  /\  F : NN --> _V )  ->  ( 2nd  o.  F
)  Fn  NN )
8482, 65, 83sylancr 646 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2nd  o.  F
)  Fn  NN )
85 fnfun 5544 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  Fun  ( 2nd  o.  F ) )
8684, 85syl 16 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 2nd  o.  F ) )
87 fndm 5546 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  dom  ( 2nd  o.  F )  =  NN )
88 eqimss2 3403 . . . . . . . . . . . 12  |-  ( dom  ( 2nd  o.  F
)  =  NN  ->  NN  C_  dom  ( 2nd  o.  F ) )
8984, 87, 883syl 19 . . . . . . . . . . 11  |-  ( ph  ->  NN  C_  dom  ( 2nd 
o.  F ) )
90 dfimafn2 5778 . . . . . . . . . . 11  |-  ( ( Fun  ( 2nd  o.  F )  /\  NN  C_ 
dom  ( 2nd  o.  F ) )  -> 
( ( 2nd  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 2nd  o.  F ) `
 x ) } )
9186, 89, 90syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 2nd  o.  F ) `
 x ) } )
92 fnima 5565 . . . . . . . . . . 11  |-  ( ( 2nd  o.  F )  Fn  NN  ->  (
( 2nd  o.  F
) " NN )  =  ran  ( 2nd 
o.  F ) )
9384, 92syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  o.  F ) " NN )  =  ran  ( 2nd 
o.  F ) )
9491, 93eqtr3d 2472 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ran  ( 2nd  o.  F ) )
95 rnco2 5379 . . . . . . . . 9  |-  ran  ( 2nd  o.  F )  =  ( 2nd " ran  F )
9694, 95syl6eq 2486 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ( 2nd " ran  F
) )
9779, 96uneq12d 3504 . . . . . . 7  |-  ( ph  ->  ( U_ x  e.  NN  { ( ( 1st  o.  F ) `
 x ) }  u.  U_ x  e.  NN  { ( ( 2nd  o.  F ) `
 x ) } )  =  ( ( 1st " ran  F
)  u.  ( 2nd " ran  F ) ) )
9859, 97syl5eq 2482 . . . . . 6  |-  ( ph  ->  U_ x  e.  NN  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )  =  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
9958, 98uneq12d 3504 . . . . 5  |-  ( ph  ->  ( U_ x  e.  NN  ( ( (,) 
o.  F ) `  x )  u.  U_ x  e.  NN  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } ) )  =  ( U. ran  ( (,)  o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
10051, 99syl5eq 2482 . . . 4  |-  ( ph  ->  U_ x  e.  NN  ( ( ( (,) 
o.  F ) `  x )  u.  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } ) )  =  ( U. ran  ( (,)  o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
10137, 50, 1003eqtr3d 2478 . . 3  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  =  ( U. ran  ( (,)  o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
1021, 101syl5sseqr 3399 . 2  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
103 ovolficcss 19368 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
1042, 103syl 16 . . . 4  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  RR )
105104ssdifssd 3487 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  RR )
106 omelon 7603 . . . . . . . . . . 11  |-  om  e.  On
107 nnenom 11321 . . . . . . . . . . . 12  |-  NN  ~~  om
108107ensymi 7159 . . . . . . . . . . 11  |-  om  ~~  NN
109 isnumi 7835 . . . . . . . . . . 11  |-  ( ( om  e.  On  /\  om 
~~  NN )  ->  NN  e.  dom  card )
110106, 108, 109mp2an 655 . . . . . . . . . 10  |-  NN  e.  dom  card
111 fofun 5656 . . . . . . . . . . . . 13  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
11260, 111ax-mp 8 . . . . . . . . . . . 12  |-  Fun  1st
113 ssv 3370 . . . . . . . . . . . . 13  |-  ran  F  C_ 
_V
114 fof 5655 . . . . . . . . . . . . . . 15  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
11560, 114ax-mp 8 . . . . . . . . . . . . . 14  |-  1st : _V
--> _V
116115fdmi 5598 . . . . . . . . . . . . 13  |-  dom  1st  =  _V
117113, 116sseqtr4i 3383 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  1st
118 fores 5664 . . . . . . . . . . . 12  |-  ( ( Fun  1st  /\  ran  F  C_ 
dom  1st )  ->  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F ) )
119112, 117, 118mp2an 655 . . . . . . . . . . 11  |-  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F )
120 ffn 5593 . . . . . . . . . . . . 13  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
1212, 120syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  F  Fn  NN )
122 dffn4 5661 . . . . . . . . . . . 12  |-  ( F  Fn  NN  <->  F : NN -onto-> ran  F )
123121, 122sylib 190 . . . . . . . . . . 11  |-  ( ph  ->  F : NN -onto-> ran  F )
124 foco 5665 . . . . . . . . . . 11  |-  ( ( ( 1st  |`  ran  F
) : ran  F -onto->
( 1st " ran  F )  /\  F : NN -onto-> ran  F )  -> 
( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F ) )
125119, 123, 124sylancr 646 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F ) )
126 fodomnum 7940 . . . . . . . . . 10  |-  ( NN  e.  dom  card  ->  ( ( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F )  -> 
( 1st " ran  F )  ~<_  NN ) )
127110, 125, 126mpsyl 62 . . . . . . . . 9  |-  ( ph  ->  ( 1st " ran  F )  ~<_  NN )
128 domentr 7168 . . . . . . . . 9  |-  ( ( ( 1st " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 1st " ran  F )  ~<_  om )
129127, 107, 128sylancl 645 . . . . . . . 8  |-  ( ph  ->  ( 1st " ran  F )  ~<_  om )
130 fofun 5656 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
13180, 130ax-mp 8 . . . . . . . . . . . 12  |-  Fun  2nd
132 fof 5655 . . . . . . . . . . . . . . 15  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
13380, 132ax-mp 8 . . . . . . . . . . . . . 14  |-  2nd : _V
--> _V
134133fdmi 5598 . . . . . . . . . . . . 13  |-  dom  2nd  =  _V
135113, 134sseqtr4i 3383 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  2nd
136 fores 5664 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  ran  F  C_ 
dom  2nd )  ->  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F ) )
137131, 135, 136mp2an 655 . . . . . . . . . . 11  |-  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F )
138 foco 5665 . . . . . . . . . . 11  |-  ( ( ( 2nd  |`  ran  F
) : ran  F -onto->
( 2nd " ran  F )  /\  F : NN -onto-> ran  F )  -> 
( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F ) )
139137, 123, 138sylancr 646 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F ) )
140 fodomnum 7940 . . . . . . . . . 10  |-  ( NN  e.  dom  card  ->  ( ( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F )  -> 
( 2nd " ran  F )  ~<_  NN ) )
141110, 139, 140mpsyl 62 . . . . . . . . 9  |-  ( ph  ->  ( 2nd " ran  F )  ~<_  NN )
142 domentr 7168 . . . . . . . . 9  |-  ( ( ( 2nd " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 2nd " ran  F )  ~<_  om )
143141, 107, 142sylancl 645 . . . . . . . 8  |-  ( ph  ->  ( 2nd " ran  F )  ~<_  om )
144 unctb 8087 . . . . . . . 8  |-  ( ( ( 1st " ran  F )  ~<_  om  /\  ( 2nd " ran  F )  ~<_  om )  ->  (
( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
145129, 143, 144syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
146 reldom 7117 . . . . . . . 8  |-  Rel  ~<_
147146brrelexi 4920 . . . . . . 7  |-  ( ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  e.  _V )
148145, 147syl 16 . . . . . 6  |-  ( ph  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  e.  _V )
149 ssid 3369 . . . . . . . 8  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
150149, 101syl5sseq 3398 . . . . . . 7  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  ( U. ran  ( (,) 
o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
151 ssundif 3713 . . . . . . 7  |-  ( U. ran  ( [,]  o.  F
)  C_  ( U. ran  ( (,)  o.  F
)  u.  ( ( 1st " ran  F
)  u.  ( 2nd " ran  F ) ) )  <->  ( U. ran  ( [,]  o.  F ) 
\  U. ran  ( (,) 
o.  F ) ) 
C_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
152150, 151sylib 190 . . . . . 6  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
153 ssdomg 7155 . . . . . 6  |-  ( ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  e.  _V  ->  (
( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ->  ( U. ran  ( [,]  o.  F ) 
\  U. ran  ( (,) 
o.  F ) )  ~<_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
154148, 152, 153sylc 59 . . . . 5  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
155 domtr 7162 . . . . 5  |-  ( ( ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  /\  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om )
156154, 145, 155syl2anc 644 . . . 4  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om )
157 domentr 7168 . . . 4  |-  ( ( ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om 
/\  om  ~~  NN )  ->  ( U. ran  ( [,]  o.  F ) 
\  U. ran  ( (,) 
o.  F ) )  ~<_  NN )
158156, 108, 157sylancl 645 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  NN )
159 ovolctb2 19390 . . 3  |-  ( ( ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  RR  /\  ( U. ran  ( [,]  o.  F ) 
\  U. ran  ( (,) 
o.  F ) )  ~<_  NN )  ->  ( vol * `  ( U. ran  ( [,]  o.  F
)  \  U. ran  ( (,)  o.  F ) ) )  =  0 )
160105, 158, 159syl2anc 644 . 2  |-  ( ph  ->  ( vol * `  ( U. ran  ( [,] 
o.  F )  \  U. ran  ( (,)  o.  F ) ) )  =  0 )
161102, 160jca 520 1  |-  ( ph  ->  ( U. ran  ( (,)  o.  F )  C_  U.
ran  ( [,]  o.  F )  /\  ( vol * `  ( U. ran  ( [,]  o.  F
)  \  U. ran  ( (,)  o.  F ) ) )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   {csn 3816   {cpr 3817   <.cop 3819   U.cuni 4017   U_ciun 4095   class class class wbr 4214   Oncon0 4583   omcom 4847    X. cxp 4878   dom cdm 4880   ran crn 4881    |` cres 4882   "cima 4883    o. ccom 4884   Fun wfun 5450    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350    ~~ cen 7108    ~<_ cdom 7109   cardccrd 7824   RRcr 8991   0cc0 8992   RR*cxr 9121    <_ cle 9123   NNcn 10002   (,)cioo 10918   [,]cicc 10921   vol *covol 19361
This theorem is referenced by:  uniioombllem3  19479  uniioombllem4  19480  uniioombllem5  19481  uniiccmbl  19484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-acn 7831  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xadd 10713  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-xmet 16697  df-met 16698  df-ovol 19363
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