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Theorem uniiccdif 21965
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 3652 . . 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 21856 . . . . . . . 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 471 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st `  ( F `
 x ) )  e.  RR  /\  ( 2nd `  ( F `  x ) )  e.  RR  /\  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) ) )
5 rexr 9642 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) )  e.  RR  ->  ( 1st `  ( F `  x ) )  e. 
RR* )
6 rexr 9642 . . . . . . . 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 11660 . . . . . . . 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 1271 . . . . . . 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 5935 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
122, 11sylan 471 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( (,) `  ( F `  x )
) )
13 inss2 3704 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
142ffvelrnda 6016 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1513, 14sseldi 3487 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  ( RR  X.  RR ) )
16 1st2nd2 6822 . . . . . . . . . . 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 5860 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
19 df-ov 6284 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
2018, 19syl6eqr 2502 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
2112, 20eqtrd 2484 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
22 df-pr 4017 . . . . . . . 8  |-  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )
23 fvco3 5935 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 1st  o.  F
) `  x )  =  ( 1st `  ( F `  x )
) )
242, 23sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st  o.  F ) `
 x )  =  ( 1st `  ( F `  x )
) )
25 fvco3 5935 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 2nd  o.  F
) `  x )  =  ( 2nd `  ( F `  x )
) )
262, 25sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd  o.  F ) `
 x )  =  ( 2nd `  ( F `  x )
) )
2724, 26preq12d 4102 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
2822, 27syl5eqr 2498 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( { ( ( 1st  o.  F ) `  x
) }  u.  {
( ( 2nd  o.  F ) `  x
) } )  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
2921, 28uneq12d 3644 . . . . . 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 5935 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
312, 30sylan 471 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( [,] `  ( F `  x )
) )
3217fveq2d 5860 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
33 df-ov 6284 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
3432, 33syl6eqr 2502 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3531, 34eqtrd 2484 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3610, 29, 353eqtr4rd 2495 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( ( (,) 
o.  F ) `  x )  u.  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } ) ) )
3736iuneq2dv 4337 . . . 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 11634 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
39 ffn 5721 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4038, 39ax-mp 5 . . . . . 6  |-  [,]  Fn  ( RR*  X.  RR* )
41 rexpssxrxp 9641 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
4213, 41sstri 3498 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
43 fss 5729 . . . . . . 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 662 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
45 fnfco 5740 . . . . . 6  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( [,]  o.  F )  Fn  NN )
4640, 44, 45sylancr 663 . . . . 5  |-  ( ph  ->  ( [,]  o.  F
)  Fn  NN )
47 fniunfv 6144 . . . . 5  |-  ( ( [,]  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x )  =  U. ran  ( [,]  o.  F
) )
4846, 47syl 16 . . . 4  |-  ( ph  ->  U_ x  e.  NN  ( ( [,]  o.  F ) `  x
)  =  U. ran  ( [,]  o.  F ) )
49 iunun 4396 . . . . 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 11633 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
51 ffn 5721 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
5250, 51ax-mp 5 . . . . . . . 8  |-  (,)  Fn  ( RR*  X.  RR* )
53 fnfco 5740 . . . . . . . 8  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( (,)  o.  F )  Fn  NN )
5452, 44, 53sylancr 663 . . . . . . 7  |-  ( ph  ->  ( (,)  o.  F
)  Fn  NN )
55 fniunfv 6144 . . . . . . 7  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( (,) 
o.  F ) `  x )  =  U. ran  ( (,)  o.  F
) )
5654, 55syl 16 . . . . . 6  |-  ( ph  ->  U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  =  U. ran  ( (,)  o.  F ) )
57 iunun 4396 . . . . . . 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 6805 . . . . . . . . . . . . . 14  |-  1st : _V -onto-> _V
59 fofn 5787 . . . . . . . . . . . . . 14  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
6058, 59ax-mp 5 . . . . . . . . . . . . 13  |-  1st  Fn  _V
61 ssv 3509 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  _V
62 fss 5729 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  _V )  ->  F : NN --> _V )
632, 61, 62sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> _V )
64 fnfco 5740 . . . . . . . . . . . . 13  |-  ( ( 1st  Fn  _V  /\  F : NN --> _V )  ->  ( 1st  o.  F
)  Fn  NN )
6560, 63, 64sylancr 663 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st  o.  F
)  Fn  NN )
66 fnfun 5668 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  Fun  ( 1st  o.  F ) )
6765, 66syl 16 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 1st  o.  F ) )
68 fndm 5670 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  dom  ( 1st  o.  F )  =  NN )
69 eqimss2 3542 . . . . . . . . . . . 12  |-  ( dom  ( 1st  o.  F
)  =  NN  ->  NN  C_  dom  ( 1st  o.  F ) )
7065, 68, 693syl 20 . . . . . . . . . . 11  |-  ( ph  ->  NN  C_  dom  ( 1st 
o.  F ) )
71 dfimafn2 5908 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 1st  o.  F ) `
 x ) } )
73 fnima 5689 . . . . . . . . . . 11  |-  ( ( 1st  o.  F )  Fn  NN  ->  (
( 1st  o.  F
) " NN )  =  ran  ( 1st 
o.  F ) )
7465, 73syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  o.  F ) " NN )  =  ran  ( 1st 
o.  F ) )
7572, 74eqtr3d 2486 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ran  ( 1st  o.  F ) )
76 rnco2 5504 . . . . . . . . 9  |-  ran  ( 1st  o.  F )  =  ( 1st " ran  F )
7775, 76syl6eq 2500 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ( 1st " ran  F
) )
78 fo2nd 6806 . . . . . . . . . . . . . 14  |-  2nd : _V -onto-> _V
79 fofn 5787 . . . . . . . . . . . . . 14  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
8078, 79ax-mp 5 . . . . . . . . . . . . 13  |-  2nd  Fn  _V
81 fnfco 5740 . . . . . . . . . . . . 13  |-  ( ( 2nd  Fn  _V  /\  F : NN --> _V )  ->  ( 2nd  o.  F
)  Fn  NN )
8280, 63, 81sylancr 663 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2nd  o.  F
)  Fn  NN )
83 fnfun 5668 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  Fun  ( 2nd  o.  F ) )
8482, 83syl 16 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 2nd  o.  F ) )
85 fndm 5670 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  dom  ( 2nd  o.  F )  =  NN )
86 eqimss2 3542 . . . . . . . . . . . 12  |-  ( dom  ( 2nd  o.  F
)  =  NN  ->  NN  C_  dom  ( 2nd  o.  F ) )
8782, 85, 863syl 20 . . . . . . . . . . 11  |-  ( ph  ->  NN  C_  dom  ( 2nd 
o.  F ) )
88 dfimafn2 5908 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 2nd  o.  F ) `
 x ) } )
90 fnima 5689 . . . . . . . . . . 11  |-  ( ( 2nd  o.  F )  Fn  NN  ->  (
( 2nd  o.  F
) " NN )  =  ran  ( 2nd 
o.  F ) )
9182, 90syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  o.  F ) " NN )  =  ran  ( 2nd 
o.  F ) )
9289, 91eqtr3d 2486 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ran  ( 2nd  o.  F ) )
93 rnco2 5504 . . . . . . . . 9  |-  ran  ( 2nd  o.  F )  =  ( 2nd " ran  F )
9492, 93syl6eq 2500 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ( 2nd " ran  F
) )
9577, 94uneq12d 3644 . . . . . . 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 2496 . . . . . 6  |-  ( ph  ->  U_ x  e.  NN  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )  =  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
9756, 96uneq12d 3644 . . . . 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 2496 . . . 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 2492 . . 3  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  =  ( U. ran  ( (,)  o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
1001, 99syl5sseqr 3538 . 2  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
101 ovolficcss 21859 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
1022, 101syl 16 . . . 4  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  RR )
103102ssdifssd 3627 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  RR )
104 omelon 8066 . . . . . . . . . . 11  |-  om  e.  On
105 nnenom 12072 . . . . . . . . . . . 12  |-  NN  ~~  om
106105ensymi 7567 . . . . . . . . . . 11  |-  om  ~~  NN
107 isnumi 8330 . . . . . . . . . . 11  |-  ( ( om  e.  On  /\  om 
~~  NN )  ->  NN  e.  dom  card )
108104, 106, 107mp2an 672 . . . . . . . . . 10  |-  NN  e.  dom  card
109 fofun 5786 . . . . . . . . . . . . 13  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
11058, 109ax-mp 5 . . . . . . . . . . . 12  |-  Fun  1st
111 ssv 3509 . . . . . . . . . . . . 13  |-  ran  F  C_ 
_V
112 fof 5785 . . . . . . . . . . . . . . 15  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
11358, 112ax-mp 5 . . . . . . . . . . . . . 14  |-  1st : _V
--> _V
114113fdmi 5726 . . . . . . . . . . . . 13  |-  dom  1st  =  _V
115111, 114sseqtr4i 3522 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  1st
116 fores 5794 . . . . . . . . . . . 12  |-  ( ( Fun  1st  /\  ran  F  C_ 
dom  1st )  ->  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F ) )
117110, 115, 116mp2an 672 . . . . . . . . . . 11  |-  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F )
118 ffn 5721 . . . . . . . . . . . . 13  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
1192, 118syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  F  Fn  NN )
120 dffn4 5791 . . . . . . . . . . . 12  |-  ( F  Fn  NN  <->  F : NN -onto-> ran  F )
121119, 120sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  F : NN -onto-> ran  F )
122 foco 5795 . . . . . . . . . . 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 663 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F ) )
124 fodomnum 8441 . . . . . . . . . 10  |-  ( NN  e.  dom  card  ->  ( ( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F )  -> 
( 1st " ran  F )  ~<_  NN ) )
125108, 123, 124mpsyl 63 . . . . . . . . 9  |-  ( ph  ->  ( 1st " ran  F )  ~<_  NN )
126 domentr 7576 . . . . . . . . 9  |-  ( ( ( 1st " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 1st " ran  F )  ~<_  om )
127125, 105, 126sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( 1st " ran  F )  ~<_  om )
128 fofun 5786 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
12978, 128ax-mp 5 . . . . . . . . . . . 12  |-  Fun  2nd
130 fof 5785 . . . . . . . . . . . . . . 15  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
13178, 130ax-mp 5 . . . . . . . . . . . . . 14  |-  2nd : _V
--> _V
132131fdmi 5726 . . . . . . . . . . . . 13  |-  dom  2nd  =  _V
133111, 132sseqtr4i 3522 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  2nd
134 fores 5794 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  ran  F  C_ 
dom  2nd )  ->  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F ) )
135129, 133, 134mp2an 672 . . . . . . . . . . 11  |-  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F )
136 foco 5795 . . . . . . . . . . 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 663 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F ) )
138 fodomnum 8441 . . . . . . . . . 10  |-  ( NN  e.  dom  card  ->  ( ( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F )  -> 
( 2nd " ran  F )  ~<_  NN ) )
139108, 137, 138mpsyl 63 . . . . . . . . 9  |-  ( ph  ->  ( 2nd " ran  F )  ~<_  NN )
140 domentr 7576 . . . . . . . . 9  |-  ( ( ( 2nd " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 2nd " ran  F )  ~<_  om )
141139, 105, 140sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( 2nd " ran  F )  ~<_  om )
142 unctb 8588 . . . . . . . 8  |-  ( ( ( 1st " ran  F )  ~<_  om  /\  ( 2nd " ran  F )  ~<_  om )  ->  (
( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
143127, 141, 142syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
144 reldom 7524 . . . . . . . 8  |-  Rel  ~<_
145144brrelexi 5030 . . . . . . 7  |-  ( ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  e.  _V )
146143, 145syl 16 . . . . . 6  |-  ( ph  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  e.  _V )
147 ssid 3508 . . . . . . . 8  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
148147, 99syl5sseq 3537 . . . . . . 7  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  ( U. ran  ( (,) 
o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
149 ssundif 3897 . . . . . . 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 196 . . . . . 6  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
151 ssdomg 7563 . . . . . 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 60 . . . . 5  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
153 domtr 7570 . . . . 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 661 . . . 4  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om )
155 domentr 7576 . . . 4  |-  ( ( ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om 
/\  om  ~~  NN )  ->  ( U. ran  ( [,]  o.  F ) 
\  U. ran  ( (,) 
o.  F ) )  ~<_  NN )
156154, 106, 155sylancl 662 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  NN )
157 ovolctb2 21881 . . 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 661 . 2  |-  ( ph  ->  ( vol* `  ( U. ran  ( [,] 
o.  F )  \  U. ran  ( (,)  o.  F ) ) )  =  0 )
159100, 158jca 532 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 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   ~Pcpw 3997   {csn 4014   {cpr 4016   <.cop 4020   U.cuni 4234   U_ciun 4315   class class class wbr 4437   Oncon0 4868    X. cxp 4987   dom cdm 4989   ran crn 4990    |` cres 4991   "cima 4992    o. ccom 4993   Fun wfun 5572    Fn wfn 5573   -->wf 5574   -onto->wfo 5576   ` cfv 5578  (class class class)co 6281   omcom 6685   1stc1st 6783   2ndc2nd 6784    ~~ cen 7515    ~<_ cdom 7516   cardccrd 8319   RRcr 9494   0cc0 9495   RR*cxr 9630    <_ cle 9632   NNcn 10543   (,)cioo 11540   [,]cicc 11543   vol*covol 21852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-q 11194  df-rp 11232  df-xadd 11330  df-ioo 11544  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-sum 13491  df-xmet 18391  df-met 18392  df-ovol 21854
This theorem is referenced by:  uniioombllem3  21972  uniioombllem4  21973  uniioombllem5  21974  uniiccmbl  21977
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