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Theorem uniiccdif 22522
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 3629 . . 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 22406 . . . . . . . 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 473 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st `  ( F `
 x ) )  e.  RR  /\  ( 2nd `  ( F `  x ) )  e.  RR  /\  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) ) )
5 rexr 9687 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) )  e.  RR  ->  ( 1st `  ( F `  x ) )  e. 
RR* )
6 rexr 9687 . . . . . . . 8  |-  ( ( 2nd `  ( F `
 x ) )  e.  RR  ->  ( 2nd `  ( F `  x ) )  e. 
RR* )
7 id 23 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) )  <_  ( 2nd `  ( F `  x )
)  ->  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) )
8 prunioo 11762 . . . . . . . 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 1306 . . . . . . 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 17 . . . . . 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 5955 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
122, 11sylan 473 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( (,) `  ( F `  x )
) )
13 inss2 3683 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
142ffvelrnda 6034 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1513, 14sseldi 3462 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  ( RR  X.  RR ) )
16 1st2nd2 6841 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( RR  X.  RR )  ->  ( F `
 x )  = 
<. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1715, 16syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  = 
<. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1817fveq2d 5882 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
19 df-ov 6305 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
2018, 19syl6eqr 2481 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
2112, 20eqtrd 2463 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
22 df-pr 3999 . . . . . . . 8  |-  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )
23 fvco3 5955 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 1st  o.  F
) `  x )  =  ( 1st `  ( F `  x )
) )
242, 23sylan 473 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st  o.  F ) `
 x )  =  ( 1st `  ( F `  x )
) )
25 fvco3 5955 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 2nd  o.  F
) `  x )  =  ( 2nd `  ( F `  x )
) )
262, 25sylan 473 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd  o.  F ) `
 x )  =  ( 2nd `  ( F `  x )
) )
2724, 26preq12d 4084 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
2822, 27syl5eqr 2477 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( { ( ( 1st  o.  F ) `  x
) }  u.  {
( ( 2nd  o.  F ) `  x
) } )  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
2921, 28uneq12d 3621 . . . . . 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 5955 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
312, 30sylan 473 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( [,] `  ( F `  x )
) )
3217fveq2d 5882 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
33 df-ov 6305 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
3432, 33syl6eqr 2481 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3531, 34eqtrd 2463 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3610, 29, 353eqtr4rd 2474 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( ( (,) 
o.  F ) `  x )  u.  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } ) ) )
3736iuneq2dv 4318 . . . 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 11734 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
39 ffn 5743 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4038, 39ax-mp 5 . . . . . 6  |-  [,]  Fn  ( RR*  X.  RR* )
41 rexpssxrxp 9686 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
4213, 41sstri 3473 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
43 fss 5751 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
442, 42, 43sylancl 666 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
45 fnfco 5762 . . . . . 6  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( [,]  o.  F )  Fn  NN )
4640, 44, 45sylancr 667 . . . . 5  |-  ( ph  ->  ( [,]  o.  F
)  Fn  NN )
47 fniunfv 6164 . . . . 5  |-  ( ( [,]  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x )  =  U. ran  ( [,]  o.  F
) )
4846, 47syl 17 . . . 4  |-  ( ph  ->  U_ x  e.  NN  ( ( [,]  o.  F ) `  x
)  =  U. ran  ( [,]  o.  F ) )
49 iunun 4380 . . . . 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 ) } ) )
50 ioof 11733 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
51 ffn 5743 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
5250, 51ax-mp 5 . . . . . . . 8  |-  (,)  Fn  ( RR*  X.  RR* )
53 fnfco 5762 . . . . . . . 8  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( (,)  o.  F )  Fn  NN )
5452, 44, 53sylancr 667 . . . . . . 7  |-  ( ph  ->  ( (,)  o.  F
)  Fn  NN )
55 fniunfv 6164 . . . . . . 7  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( (,) 
o.  F ) `  x )  =  U. ran  ( (,)  o.  F
) )
5654, 55syl 17 . . . . . 6  |-  ( ph  ->  U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  =  U. ran  ( (,)  o.  F ) )
57 iunun 4380 . . . . . . 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 ) } )
58 fo1st 6824 . . . . . . . . . . . . . 14  |-  1st : _V -onto-> _V
59 fofn 5809 . . . . . . . . . . . . . 14  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
6058, 59ax-mp 5 . . . . . . . . . . . . 13  |-  1st  Fn  _V
61 ssv 3484 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  _V
62 fss 5751 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  _V )  ->  F : NN --> _V )
632, 61, 62sylancl 666 . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> _V )
64 fnfco 5762 . . . . . . . . . . . . 13  |-  ( ( 1st  Fn  _V  /\  F : NN --> _V )  ->  ( 1st  o.  F
)  Fn  NN )
6560, 63, 64sylancr 667 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st  o.  F
)  Fn  NN )
66 fnfun 5688 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  Fun  ( 1st  o.  F ) )
6765, 66syl 17 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 1st  o.  F ) )
68 fndm 5690 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  dom  ( 1st  o.  F )  =  NN )
69 eqimss2 3517 . . . . . . . . . . . 12  |-  ( dom  ( 1st  o.  F
)  =  NN  ->  NN  C_  dom  ( 1st  o.  F ) )
7065, 68, 693syl 18 . . . . . . . . . . 11  |-  ( ph  ->  NN  C_  dom  ( 1st 
o.  F ) )
71 dfimafn2 5928 . . . . . . . . . . 11  |-  ( ( Fun  ( 1st  o.  F )  /\  NN  C_ 
dom  ( 1st  o.  F ) )  -> 
( ( 1st  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 1st  o.  F ) `
 x ) } )
7267, 70, 71syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 1st  o.  F ) `
 x ) } )
73 fnima 5709 . . . . . . . . . . 11  |-  ( ( 1st  o.  F )  Fn  NN  ->  (
( 1st  o.  F
) " NN )  =  ran  ( 1st 
o.  F ) )
7465, 73syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  o.  F ) " NN )  =  ran  ( 1st 
o.  F ) )
7572, 74eqtr3d 2465 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ran  ( 1st  o.  F ) )
76 rnco2 5358 . . . . . . . . 9  |-  ran  ( 1st  o.  F )  =  ( 1st " ran  F )
7775, 76syl6eq 2479 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ( 1st " ran  F
) )
78 fo2nd 6825 . . . . . . . . . . . . . 14  |-  2nd : _V -onto-> _V
79 fofn 5809 . . . . . . . . . . . . . 14  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
8078, 79ax-mp 5 . . . . . . . . . . . . 13  |-  2nd  Fn  _V
81 fnfco 5762 . . . . . . . . . . . . 13  |-  ( ( 2nd  Fn  _V  /\  F : NN --> _V )  ->  ( 2nd  o.  F
)  Fn  NN )
8280, 63, 81sylancr 667 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2nd  o.  F
)  Fn  NN )
83 fnfun 5688 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  Fun  ( 2nd  o.  F ) )
8482, 83syl 17 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 2nd  o.  F ) )
85 fndm 5690 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  dom  ( 2nd  o.  F )  =  NN )
86 eqimss2 3517 . . . . . . . . . . . 12  |-  ( dom  ( 2nd  o.  F
)  =  NN  ->  NN  C_  dom  ( 2nd  o.  F ) )
8782, 85, 863syl 18 . . . . . . . . . . 11  |-  ( ph  ->  NN  C_  dom  ( 2nd 
o.  F ) )
88 dfimafn2 5928 . . . . . . . . . . 11  |-  ( ( Fun  ( 2nd  o.  F )  /\  NN  C_ 
dom  ( 2nd  o.  F ) )  -> 
( ( 2nd  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 2nd  o.  F ) `
 x ) } )
8984, 87, 88syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 2nd  o.  F ) `
 x ) } )
90 fnima 5709 . . . . . . . . . . 11  |-  ( ( 2nd  o.  F )  Fn  NN  ->  (
( 2nd  o.  F
) " NN )  =  ran  ( 2nd 
o.  F ) )
9182, 90syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  o.  F ) " NN )  =  ran  ( 2nd 
o.  F ) )
9289, 91eqtr3d 2465 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ran  ( 2nd  o.  F ) )
93 rnco2 5358 . . . . . . . . 9  |-  ran  ( 2nd  o.  F )  =  ( 2nd " ran  F )
9492, 93syl6eq 2479 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ( 2nd " ran  F
) )
9577, 94uneq12d 3621 . . . . . . 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 ) ) )
9657, 95syl5eq 2475 . . . . . 6  |-  ( ph  ->  U_ x  e.  NN  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )  =  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
9756, 96uneq12d 3621 . . . . 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 ) ) ) )
9849, 97syl5eq 2475 . . . 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 ) ) ) )
9937, 48, 983eqtr3d 2471 . . 3  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  =  ( U. ran  ( (,)  o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
1001, 99syl5sseqr 3513 . 2  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
101 ovolficcss 22409 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
1022, 101syl 17 . . . 4  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  RR )
103102ssdifssd 3603 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  RR )
104 omelon 8154 . . . . . . . . . . 11  |-  om  e.  On
105 nnenom 12193 . . . . . . . . . . . 12  |-  NN  ~~  om
106105ensymi 7623 . . . . . . . . . . 11  |-  om  ~~  NN
107 isnumi 8382 . . . . . . . . . . 11  |-  ( ( om  e.  On  /\  om 
~~  NN )  ->  NN  e.  dom  card )
108104, 106, 107mp2an 676 . . . . . . . . . 10  |-  NN  e.  dom  card
109 fofun 5808 . . . . . . . . . . . . 13  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
11058, 109ax-mp 5 . . . . . . . . . . . 12  |-  Fun  1st
111 ssv 3484 . . . . . . . . . . . . 13  |-  ran  F  C_ 
_V
112 fof 5807 . . . . . . . . . . . . . . 15  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
11358, 112ax-mp 5 . . . . . . . . . . . . . 14  |-  1st : _V
--> _V
114113fdmi 5748 . . . . . . . . . . . . 13  |-  dom  1st  =  _V
115111, 114sseqtr4i 3497 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  1st
116 fores 5816 . . . . . . . . . . . 12  |-  ( ( Fun  1st  /\  ran  F  C_ 
dom  1st )  ->  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F ) )
117110, 115, 116mp2an 676 . . . . . . . . . . 11  |-  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F )
118 ffn 5743 . . . . . . . . . . . . 13  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
1192, 118syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  F  Fn  NN )
120 dffn4 5813 . . . . . . . . . . . 12  |-  ( F  Fn  NN  <->  F : NN -onto-> ran  F )
121119, 120sylib 199 . . . . . . . . . . 11  |-  ( ph  ->  F : NN -onto-> ran  F )
122 foco 5817 . . . . . . . . . . 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 ) )
123117, 121, 122sylancr 667 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F ) )
124 fodomnum 8489 . . . . . . . . . 10  |-  ( NN  e.  dom  card  ->  ( ( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F )  -> 
( 1st " ran  F )  ~<_  NN ) )
125108, 123, 124mpsyl 65 . . . . . . . . 9  |-  ( ph  ->  ( 1st " ran  F )  ~<_  NN )
126 domentr 7632 . . . . . . . . 9  |-  ( ( ( 1st " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 1st " ran  F )  ~<_  om )
127125, 105, 126sylancl 666 . . . . . . . 8  |-  ( ph  ->  ( 1st " ran  F )  ~<_  om )
128 fofun 5808 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
12978, 128ax-mp 5 . . . . . . . . . . . 12  |-  Fun  2nd
130 fof 5807 . . . . . . . . . . . . . . 15  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
13178, 130ax-mp 5 . . . . . . . . . . . . . 14  |-  2nd : _V
--> _V
132131fdmi 5748 . . . . . . . . . . . . 13  |-  dom  2nd  =  _V
133111, 132sseqtr4i 3497 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  2nd
134 fores 5816 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  ran  F  C_ 
dom  2nd )  ->  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F ) )
135129, 133, 134mp2an 676 . . . . . . . . . . 11  |-  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F )
136 foco 5817 . . . . . . . . . . 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 ) )
137135, 121, 136sylancr 667 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F ) )
138 fodomnum 8489 . . . . . . . . . 10  |-  ( NN  e.  dom  card  ->  ( ( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F )  -> 
( 2nd " ran  F )  ~<_  NN ) )
139108, 137, 138mpsyl 65 . . . . . . . . 9  |-  ( ph  ->  ( 2nd " ran  F )  ~<_  NN )
140 domentr 7632 . . . . . . . . 9  |-  ( ( ( 2nd " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 2nd " ran  F )  ~<_  om )
141139, 105, 140sylancl 666 . . . . . . . 8  |-  ( ph  ->  ( 2nd " ran  F )  ~<_  om )
142 unctb 8636 . . . . . . . 8  |-  ( ( ( 1st " ran  F )  ~<_  om  /\  ( 2nd " ran  F )  ~<_  om )  ->  (
( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
143127, 141, 142syl2anc 665 . . . . . . 7  |-  ( ph  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
144 reldom 7580 . . . . . . . 8  |-  Rel  ~<_
145144brrelexi 4891 . . . . . . 7  |-  ( ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  e.  _V )
146143, 145syl 17 . . . . . 6  |-  ( ph  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  e.  _V )
147 ssid 3483 . . . . . . . 8  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
148147, 99syl5sseq 3512 . . . . . . 7  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  ( U. ran  ( (,) 
o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
149 ssundif 3879 . . . . . . 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 ) ) )
150148, 149sylib 199 . . . . . 6  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
151 ssdomg 7619 . . . . . 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 ) ) ) )
152146, 150, 151sylc 62 . . . . 5  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
153 domtr 7626 . . . . 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 )
154152, 143, 153syl2anc 665 . . . 4  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om )
155 domentr 7632 . . . 4  |-  ( ( ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om 
/\  om  ~~  NN )  ->  ( U. ran  ( [,]  o.  F ) 
\  U. ran  ( (,) 
o.  F ) )  ~<_  NN )
156154, 106, 155sylancl 666 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  NN )
157 ovolctb2 22432 . . 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 )
158103, 156, 157syl2anc 665 . 2  |-  ( ph  ->  ( vol* `  ( U. ran  ( [,] 
o.  F )  \  U. ran  ( (,)  o.  F ) ) )  =  0 )
159100, 158jca 534 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   _Vcvv 3081    \ cdif 3433    u. cun 3434    i^i cin 3435    C_ wss 3436   ~Pcpw 3979   {csn 3996   {cpr 3998   <.cop 4002   U.cuni 4216   U_ciun 4296   class class class wbr 4420    X. cxp 4848   dom cdm 4850   ran crn 4851    |` cres 4852   "cima 4853    o. ccom 4854   Oncon0 5439   Fun wfun 5592    Fn wfn 5593   -->wf 5594   -onto->wfo 5596   ` cfv 5598  (class class class)co 6302   omcom 6703   1stc1st 6802   2ndc2nd 6803    ~~ cen 7571    ~<_ cdom 7572   cardccrd 8371   RRcr 9539   0cc0 9540   RR*cxr 9675    <_ cle 9677   NNcn 10610   (,)cioo 11636   [,]cicc 11639   vol*covol 22400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-acn 8378  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-q 11266  df-rp 11304  df-xadd 11411  df-ioo 11640  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-xmet 18951  df-met 18952  df-ovol 22403
This theorem is referenced by:  uniioombllem3  22530  uniioombllem4  22531  uniioombllem5  22532  uniiccmbl  22535
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