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Theorem uniiccdif 22583
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 3608 . . 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 22467 . . . . . . . 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 478 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st `  ( F `
 x ) )  e.  RR  /\  ( 2nd `  ( F `  x ) )  e.  RR  /\  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) ) )
5 rexr 9711 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) )  e.  RR  ->  ( 1st `  ( F `  x ) )  e. 
RR* )
6 rexr 9711 . . . . . . . 8  |-  ( ( 2nd `  ( F `
 x ) )  e.  RR  ->  ( 2nd `  ( F `  x ) )  e. 
RR* )
7 id 22 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) )  <_  ( 2nd `  ( F `  x )
)  ->  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) )
8 prunioo 11789 . . . . . . . 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 1318 . . . . . . 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 5964 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
122, 11sylan 478 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( (,) `  ( F `  x )
) )
13 inss2 3664 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
142ffvelrnda 6044 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1513, 14sseldi 3441 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  ( RR  X.  RR ) )
16 1st2nd2 6856 . . . . . . . . . . 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 5891 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
19 df-ov 6317 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
2018, 19syl6eqr 2513 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
2112, 20eqtrd 2495 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
22 df-pr 3982 . . . . . . . 8  |-  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )
23 fvco3 5964 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 1st  o.  F
) `  x )  =  ( 1st `  ( F `  x )
) )
242, 23sylan 478 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st  o.  F ) `
 x )  =  ( 1st `  ( F `  x )
) )
25 fvco3 5964 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 2nd  o.  F
) `  x )  =  ( 2nd `  ( F `  x )
) )
262, 25sylan 478 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd  o.  F ) `
 x )  =  ( 2nd `  ( F `  x )
) )
2724, 26preq12d 4071 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
2822, 27syl5eqr 2509 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( { ( ( 1st  o.  F ) `  x
) }  u.  {
( ( 2nd  o.  F ) `  x
) } )  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
2921, 28uneq12d 3600 . . . . . 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 5964 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
312, 30sylan 478 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( [,] `  ( F `  x )
) )
3217fveq2d 5891 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
33 df-ov 6317 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
3432, 33syl6eqr 2513 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3531, 34eqtrd 2495 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3610, 29, 353eqtr4rd 2506 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( ( (,) 
o.  F ) `  x )  u.  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } ) ) )
3736iuneq2dv 4313 . . . 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 11761 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
39 ffn 5750 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4038, 39ax-mp 5 . . . . . 6  |-  [,]  Fn  ( RR*  X.  RR* )
41 rexpssxrxp 9710 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
4213, 41sstri 3452 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
43 fss 5759 . . . . . . 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 673 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
45 fnfco 5770 . . . . . 6  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( [,]  o.  F )  Fn  NN )
4640, 44, 45sylancr 674 . . . . 5  |-  ( ph  ->  ( [,]  o.  F
)  Fn  NN )
47 fniunfv 6176 . . . . 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 4375 . . . . 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 11760 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
51 ffn 5750 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
5250, 51ax-mp 5 . . . . . . . 8  |-  (,)  Fn  ( RR*  X.  RR* )
53 fnfco 5770 . . . . . . . 8  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( (,)  o.  F )  Fn  NN )
5452, 44, 53sylancr 674 . . . . . . 7  |-  ( ph  ->  ( (,)  o.  F
)  Fn  NN )
55 fniunfv 6176 . . . . . . 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 4375 . . . . . . 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 6839 . . . . . . . . . . . . . 14  |-  1st : _V -onto-> _V
59 fofn 5817 . . . . . . . . . . . . . 14  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
6058, 59ax-mp 5 . . . . . . . . . . . . 13  |-  1st  Fn  _V
61 ssv 3463 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  _V
62 fss 5759 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  _V )  ->  F : NN --> _V )
632, 61, 62sylancl 673 . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> _V )
64 fnfco 5770 . . . . . . . . . . . . 13  |-  ( ( 1st  Fn  _V  /\  F : NN --> _V )  ->  ( 1st  o.  F
)  Fn  NN )
6560, 63, 64sylancr 674 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st  o.  F
)  Fn  NN )
66 fnfun 5694 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  Fun  ( 1st  o.  F ) )
6765, 66syl 17 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 1st  o.  F ) )
68 fndm 5696 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  dom  ( 1st  o.  F )  =  NN )
69 eqimss2 3496 . . . . . . . . . . . 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 5937 . . . . . . . . . . 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 671 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 1st  o.  F ) `
 x ) } )
73 fnima 5715 . . . . . . . . . . 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 2497 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ran  ( 1st  o.  F ) )
76 rnco2 5360 . . . . . . . . 9  |-  ran  ( 1st  o.  F )  =  ( 1st " ran  F )
7775, 76syl6eq 2511 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ( 1st " ran  F
) )
78 fo2nd 6840 . . . . . . . . . . . . . 14  |-  2nd : _V -onto-> _V
79 fofn 5817 . . . . . . . . . . . . . 14  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
8078, 79ax-mp 5 . . . . . . . . . . . . 13  |-  2nd  Fn  _V
81 fnfco 5770 . . . . . . . . . . . . 13  |-  ( ( 2nd  Fn  _V  /\  F : NN --> _V )  ->  ( 2nd  o.  F
)  Fn  NN )
8280, 63, 81sylancr 674 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2nd  o.  F
)  Fn  NN )
83 fnfun 5694 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  Fun  ( 2nd  o.  F ) )
8482, 83syl 17 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 2nd  o.  F ) )
85 fndm 5696 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  dom  ( 2nd  o.  F )  =  NN )
86 eqimss2 3496 . . . . . . . . . . . 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 5937 . . . . . . . . . . 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 671 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 2nd  o.  F ) `
 x ) } )
90 fnima 5715 . . . . . . . . . . 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 2497 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ran  ( 2nd  o.  F ) )
93 rnco2 5360 . . . . . . . . 9  |-  ran  ( 2nd  o.  F )  =  ( 2nd " ran  F )
9492, 93syl6eq 2511 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ( 2nd " ran  F
) )
9577, 94uneq12d 3600 . . . . . . 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 2507 . . . . . 6  |-  ( ph  ->  U_ x  e.  NN  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )  =  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
9756, 96uneq12d 3600 . . . . 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 2507 . . . 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 2503 . . 3  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  =  ( U. ran  ( (,)  o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
1001, 99syl5sseqr 3492 . 2  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
101 ovolficcss 22470 . . . . 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 3582 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  RR )
104 omelon 8176 . . . . . . . . . . 11  |-  om  e.  On
105 nnenom 12224 . . . . . . . . . . . 12  |-  NN  ~~  om
106105ensymi 7644 . . . . . . . . . . 11  |-  om  ~~  NN
107 isnumi 8405 . . . . . . . . . . 11  |-  ( ( om  e.  On  /\  om 
~~  NN )  ->  NN  e.  dom  card )
108104, 106, 107mp2an 683 . . . . . . . . . 10  |-  NN  e.  dom  card
109 fofun 5816 . . . . . . . . . . . . 13  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
11058, 109ax-mp 5 . . . . . . . . . . . 12  |-  Fun  1st
111 ssv 3463 . . . . . . . . . . . . 13  |-  ran  F  C_ 
_V
112 fof 5815 . . . . . . . . . . . . . . 15  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
11358, 112ax-mp 5 . . . . . . . . . . . . . 14  |-  1st : _V
--> _V
114113fdmi 5756 . . . . . . . . . . . . 13  |-  dom  1st  =  _V
115111, 114sseqtr4i 3476 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  1st
116 fores 5824 . . . . . . . . . . . 12  |-  ( ( Fun  1st  /\  ran  F  C_ 
dom  1st )  ->  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F ) )
117110, 115, 116mp2an 683 . . . . . . . . . . 11  |-  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F )
118 ffn 5750 . . . . . . . . . . . . 13  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
1192, 118syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  F  Fn  NN )
120 dffn4 5821 . . . . . . . . . . . 12  |-  ( F  Fn  NN  <->  F : NN -onto-> ran  F )
121119, 120sylib 201 . . . . . . . . . . 11  |-  ( ph  ->  F : NN -onto-> ran  F )
122 foco 5825 . . . . . . . . . . 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 674 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F ) )
124 fodomnum 8513 . . . . . . . . . 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 7653 . . . . . . . . 9  |-  ( ( ( 1st " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 1st " ran  F )  ~<_  om )
127125, 105, 126sylancl 673 . . . . . . . 8  |-  ( ph  ->  ( 1st " ran  F )  ~<_  om )
128 fofun 5816 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
12978, 128ax-mp 5 . . . . . . . . . . . 12  |-  Fun  2nd
130 fof 5815 . . . . . . . . . . . . . . 15  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
13178, 130ax-mp 5 . . . . . . . . . . . . . 14  |-  2nd : _V
--> _V
132131fdmi 5756 . . . . . . . . . . . . 13  |-  dom  2nd  =  _V
133111, 132sseqtr4i 3476 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  2nd
134 fores 5824 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  ran  F  C_ 
dom  2nd )  ->  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F ) )
135129, 133, 134mp2an 683 . . . . . . . . . . 11  |-  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F )
136 foco 5825 . . . . . . . . . . 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 674 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F ) )
138 fodomnum 8513 . . . . . . . . . 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 7653 . . . . . . . . 9  |-  ( ( ( 2nd " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 2nd " ran  F )  ~<_  om )
141139, 105, 140sylancl 673 . . . . . . . 8  |-  ( ph  ->  ( 2nd " ran  F )  ~<_  om )
142 unctb 8660 . . . . . . . 8  |-  ( ( ( 1st " ran  F )  ~<_  om  /\  ( 2nd " ran  F )  ~<_  om )  ->  (
( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
143127, 141, 142syl2anc 671 . . . . . . 7  |-  ( ph  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
144 reldom 7600 . . . . . . . 8  |-  Rel  ~<_
145144brrelexi 4893 . . . . . . 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 3462 . . . . . . . 8  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
148147, 99syl5sseq 3491 . . . . . . 7  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  ( U. ran  ( (,) 
o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
149 ssundif 3862 . . . . . . 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 201 . . . . . 6  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
151 ssdomg 7640 . . . . . 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 7647 . . . . 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 671 . . . 4  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om )
155 domentr 7653 . . . 4  |-  ( ( ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om 
/\  om  ~~  NN )  ->  ( U. ran  ( [,]  o.  F ) 
\  U. ran  ( (,) 
o.  F ) )  ~<_  NN )
156154, 106, 155sylancl 673 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  NN )
157 ovolctb2 22493 . . 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 671 . 2  |-  ( ph  ->  ( vol* `  ( U. ran  ( [,] 
o.  F )  \  U. ran  ( (,)  o.  F ) ) )  =  0 )
159100, 158jca 539 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 375    /\ w3a 991    = wceq 1454    e. wcel 1897   _Vcvv 3056    \ cdif 3412    u. cun 3413    i^i cin 3414    C_ wss 3415   ~Pcpw 3962   {csn 3979   {cpr 3981   <.cop 3985   U.cuni 4211   U_ciun 4291   class class class wbr 4415    X. cxp 4850   dom cdm 4852   ran crn 4853    |` cres 4854   "cima 4855    o. ccom 4856   Oncon0 5441   Fun wfun 5594    Fn wfn 5595   -->wf 5596   -onto->wfo 5598   ` cfv 5600  (class class class)co 6314   omcom 6718   1stc1st 6817   2ndc2nd 6818    ~~ cen 7591    ~<_ cdom 7592   cardccrd 8394   RRcr 9563   0cc0 9564   RR*cxr 9699    <_ cle 9701   NNcn 10636   (,)cioo 11663   [,]cicc 11666   vol*covol 22461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-2o 7208  df-oadd 7211  df-er 7388  df-map 7499  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-sup 7981  df-inf 7982  df-oi 8050  df-card 8398  df-acn 8401  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-q 11293  df-rp 11331  df-xadd 11438  df-ioo 11667  df-ico 11669  df-icc 11670  df-fz 11813  df-fzo 11946  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-clim 13600  df-sum 13801  df-xmet 19011  df-met 19012  df-ovol 22464
This theorem is referenced by:  uniioombllem3  22591  uniioombllem4  22592  uniioombllem5  22593  uniiccmbl  22596
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