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Theorem uniiccdif 18933
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 3338 . . 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 18826 . . . . . . . 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 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st `  ( F `
 x ) )  e.  RR  /\  ( 2nd `  ( F `  x ) )  e.  RR  /\  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) ) )
5 rexr 8877 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) )  e.  RR  ->  ( 1st `  ( F `  x ) )  e. 
RR* )
6 rexr 8877 . . . . . . . 8  |-  ( ( 2nd `  ( F `
 x ) )  e.  RR  ->  ( 2nd `  ( F `  x ) )  e. 
RR* )
7 id 19 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) )  <_  ( 2nd `  ( F `  x )
)  ->  ( 1st `  ( F `  x
) )  <_  ( 2nd `  ( F `  x ) ) )
8 prunioo 10764 . . . . . . . 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 1224 . . . . . . 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 15 . . . . . 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 5596 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( (,)  o.  F
) `  x )  =  ( (,) `  ( F `  x )
) )
122, 11sylan 457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( (,) `  ( F `  x )
) )
13 inss2 3390 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
14 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  ( F `  x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
152, 14sylan 457 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1613, 15sseldi 3178 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  ( RR  X.  RR ) )
17 1st2nd2 6159 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( RR  X.  RR )  ->  ( F `
 x )  = 
<. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1816, 17syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  = 
<. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
1918fveq2d 5529 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
20 df-ov 5861 . . . . . . . . 9  |-  ( ( 1st `  ( F `
 x ) ) (,) ( 2nd `  ( F `  x )
) )  =  ( (,) `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
2119, 20syl6eqr 2333 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( (,) `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
2212, 21eqtrd 2315 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( (,)  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) (,) ( 2nd `  ( F `  x
) ) ) )
23 df-pr 3647 . . . . . . . 8  |-  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )
24 fvco3 5596 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 1st  o.  F
) `  x )  =  ( 1st `  ( F `  x )
) )
252, 24sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 1st  o.  F ) `
 x )  =  ( 1st `  ( F `  x )
) )
26 fvco3 5596 . . . . . . . . . 10  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( 2nd  o.  F
) `  x )  =  ( 2nd `  ( F `  x )
) )
272, 26sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd  o.  F ) `
 x )  =  ( 2nd `  ( F `  x )
) )
2825, 27preq12d 3714 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  { ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) }  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
2923, 28syl5eqr 2329 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( { ( ( 1st  o.  F ) `  x
) }  u.  {
( ( 2nd  o.  F ) `  x
) } )  =  { ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) } )
3022, 29uneq12d 3330 . . . . . 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
) ) } ) )
31 fvco3 5596 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  NN )  ->  (
( [,]  o.  F
) `  x )  =  ( [,] `  ( F `  x )
) )
322, 31sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( [,] `  ( F `  x )
) )
3318fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
34 df-ov 5861 . . . . . . . 8  |-  ( ( 1st `  ( F `
 x ) ) [,] ( 2nd `  ( F `  x )
) )  =  ( [,] `  <. ( 1st `  ( F `  x ) ) ,  ( 2nd `  ( F `  x )
) >. )
3533, 34syl6eqr 2333 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  ( [,] `  ( F `  x
) )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3632, 35eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( 1st `  ( F `  x )
) [,] ( 2nd `  ( F `  x
) ) ) )
3710, 30, 363eqtr4rd 2326 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( [,]  o.  F ) `
 x )  =  ( ( ( (,) 
o.  F ) `  x )  u.  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } ) ) )
3837iuneq2dv 3926 . . . 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 ) } ) ) )
39 iccf 10742 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
40 ffn 5389 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
4139, 40ax-mp 8 . . . . . 6  |-  [,]  Fn  ( RR*  X.  RR* )
42 ressxr 8876 . . . . . . . . 9  |-  RR  C_  RR*
43 xpss12 4792 . . . . . . . . 9  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
4442, 42, 43mp2an 653 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
4513, 44sstri 3188 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
46 fss 5397 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
472, 45, 46sylancl 643 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR*  X. 
RR* ) )
48 fnfco 5407 . . . . . 6  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( [,]  o.  F )  Fn  NN )
4941, 47, 48sylancr 644 . . . . 5  |-  ( ph  ->  ( [,]  o.  F
)  Fn  NN )
50 fniunfv 5773 . . . . 5  |-  ( ( [,]  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( [,] 
o.  F ) `  x )  =  U. ran  ( [,]  o.  F
) )
5149, 50syl 15 . . . 4  |-  ( ph  ->  U_ x  e.  NN  ( ( [,]  o.  F ) `  x
)  =  U. ran  ( [,]  o.  F ) )
52 iunun 3982 . . . . 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 ) } ) )
53 ioof 10741 . . . . . . . . 9  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
54 ffn 5389 . . . . . . . . 9  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
5553, 54ax-mp 8 . . . . . . . 8  |-  (,)  Fn  ( RR*  X.  RR* )
56 fnfco 5407 . . . . . . . 8  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  F : NN --> ( RR*  X.  RR* ) )  ->  ( (,)  o.  F )  Fn  NN )
5755, 47, 56sylancr 644 . . . . . . 7  |-  ( ph  ->  ( (,)  o.  F
)  Fn  NN )
58 fniunfv 5773 . . . . . . 7  |-  ( ( (,)  o.  F )  Fn  NN  ->  U_ x  e.  NN  ( ( (,) 
o.  F ) `  x )  =  U. ran  ( (,)  o.  F
) )
5957, 58syl 15 . . . . . 6  |-  ( ph  ->  U_ x  e.  NN  ( ( (,)  o.  F ) `  x
)  =  U. ran  ( (,)  o.  F ) )
60 iunun 3982 . . . . . . 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 ) } )
61 fo1st 6139 . . . . . . . . . . . . . 14  |-  1st : _V -onto-> _V
62 fofn 5453 . . . . . . . . . . . . . 14  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
6361, 62ax-mp 8 . . . . . . . . . . . . 13  |-  1st  Fn  _V
64 ssv 3198 . . . . . . . . . . . . . 14  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  _V
65 fss 5397 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  _V )  ->  F : NN --> _V )
662, 64, 65sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> _V )
67 fnfco 5407 . . . . . . . . . . . . 13  |-  ( ( 1st  Fn  _V  /\  F : NN --> _V )  ->  ( 1st  o.  F
)  Fn  NN )
6863, 66, 67sylancr 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st  o.  F
)  Fn  NN )
69 fnfun 5341 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  Fun  ( 1st  o.  F ) )
7068, 69syl 15 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 1st  o.  F ) )
71 fndm 5343 . . . . . . . . . . . 12  |-  ( ( 1st  o.  F )  Fn  NN  ->  dom  ( 1st  o.  F )  =  NN )
72 eqimss2 3231 . . . . . . . . . . . 12  |-  ( dom  ( 1st  o.  F
)  =  NN  ->  NN  C_  dom  ( 1st  o.  F ) )
7368, 71, 723syl 18 . . . . . . . . . . 11  |-  ( ph  ->  NN  C_  dom  ( 1st 
o.  F ) )
74 dfimafn2 5572 . . . . . . . . . . 11  |-  ( ( Fun  ( 1st  o.  F )  /\  NN  C_ 
dom  ( 1st  o.  F ) )  -> 
( ( 1st  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 1st  o.  F ) `
 x ) } )
7570, 73, 74syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 1st  o.  F ) `
 x ) } )
76 fnima 5362 . . . . . . . . . . 11  |-  ( ( 1st  o.  F )  Fn  NN  ->  (
( 1st  o.  F
) " NN )  =  ran  ( 1st 
o.  F ) )
7768, 76syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  o.  F ) " NN )  =  ran  ( 1st 
o.  F ) )
7875, 77eqtr3d 2317 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ran  ( 1st  o.  F ) )
79 rnco2 5180 . . . . . . . . 9  |-  ran  ( 1st  o.  F )  =  ( 1st " ran  F )
8078, 79syl6eq 2331 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 1st  o.  F ) `  x
) }  =  ( 1st " ran  F
) )
81 fo2nd 6140 . . . . . . . . . . . . . 14  |-  2nd : _V -onto-> _V
82 fofn 5453 . . . . . . . . . . . . . 14  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
8381, 82ax-mp 8 . . . . . . . . . . . . 13  |-  2nd  Fn  _V
84 fnfco 5407 . . . . . . . . . . . . 13  |-  ( ( 2nd  Fn  _V  /\  F : NN --> _V )  ->  ( 2nd  o.  F
)  Fn  NN )
8583, 66, 84sylancr 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2nd  o.  F
)  Fn  NN )
86 fnfun 5341 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  Fun  ( 2nd  o.  F ) )
8785, 86syl 15 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( 2nd  o.  F ) )
88 fndm 5343 . . . . . . . . . . . 12  |-  ( ( 2nd  o.  F )  Fn  NN  ->  dom  ( 2nd  o.  F )  =  NN )
89 eqimss2 3231 . . . . . . . . . . . 12  |-  ( dom  ( 2nd  o.  F
)  =  NN  ->  NN  C_  dom  ( 2nd  o.  F ) )
9085, 88, 893syl 18 . . . . . . . . . . 11  |-  ( ph  ->  NN  C_  dom  ( 2nd 
o.  F ) )
91 dfimafn2 5572 . . . . . . . . . . 11  |-  ( ( Fun  ( 2nd  o.  F )  /\  NN  C_ 
dom  ( 2nd  o.  F ) )  -> 
( ( 2nd  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 2nd  o.  F ) `
 x ) } )
9287, 90, 91syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  o.  F ) " NN )  =  U_ x  e.  NN  { ( ( 2nd  o.  F ) `
 x ) } )
93 fnima 5362 . . . . . . . . . . 11  |-  ( ( 2nd  o.  F )  Fn  NN  ->  (
( 2nd  o.  F
) " NN )  =  ran  ( 2nd 
o.  F ) )
9485, 93syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  o.  F ) " NN )  =  ran  ( 2nd 
o.  F ) )
9592, 94eqtr3d 2317 . . . . . . . . 9  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ran  ( 2nd  o.  F ) )
96 rnco2 5180 . . . . . . . . 9  |-  ran  ( 2nd  o.  F )  =  ( 2nd " ran  F )
9795, 96syl6eq 2331 . . . . . . . 8  |-  ( ph  ->  U_ x  e.  NN  { ( ( 2nd  o.  F ) `  x
) }  =  ( 2nd " ran  F
) )
9880, 97uneq12d 3330 . . . . . . 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 ) ) )
9960, 98syl5eq 2327 . . . . . 6  |-  ( ph  ->  U_ x  e.  NN  ( { ( ( 1st 
o.  F ) `  x ) }  u.  { ( ( 2nd  o.  F ) `  x
) } )  =  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
10059, 99uneq12d 3330 . . . . 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 ) ) ) )
10152, 100syl5eq 2327 . . . 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 ) ) ) )
10238, 51, 1013eqtr3d 2323 . . 3  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  =  ( U. ran  ( (,)  o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
1031, 102syl5sseqr 3227 . 2  |-  ( ph  ->  U. ran  ( (,) 
o.  F )  C_  U.
ran  ( [,]  o.  F ) )
104 difss 3303 . . . 4  |-  ( U. ran  ( [,]  o.  F
)  \  U. ran  ( (,)  o.  F ) ) 
C_  U. ran  ( [,] 
o.  F )
105 ovolficcss 18829 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  U. ran  ( [,]  o.  F ) 
C_  RR )
1062, 105syl 15 . . . 4  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  RR )
107104, 106syl5ss 3190 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  RR )
108 omelon 7347 . . . . . . . . . . 11  |-  om  e.  On
109 nnenom 11042 . . . . . . . . . . . 12  |-  NN  ~~  om
110109ensymi 6911 . . . . . . . . . . 11  |-  om  ~~  NN
111 isnumi 7579 . . . . . . . . . . 11  |-  ( ( om  e.  On  /\  om 
~~  NN )  ->  NN  e.  dom  card )
112108, 110, 111mp2an 653 . . . . . . . . . 10  |-  NN  e.  dom  card
113 fofun 5452 . . . . . . . . . . . . 13  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
11461, 113ax-mp 8 . . . . . . . . . . . 12  |-  Fun  1st
115 ssv 3198 . . . . . . . . . . . . 13  |-  ran  F  C_ 
_V
116 fof 5451 . . . . . . . . . . . . . . 15  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
11761, 116ax-mp 8 . . . . . . . . . . . . . 14  |-  1st : _V
--> _V
118117fdmi 5394 . . . . . . . . . . . . 13  |-  dom  1st  =  _V
119115, 118sseqtr4i 3211 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  1st
120 fores 5460 . . . . . . . . . . . 12  |-  ( ( Fun  1st  /\  ran  F  C_ 
dom  1st )  ->  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F ) )
121114, 119, 120mp2an 653 . . . . . . . . . . 11  |-  ( 1st  |`  ran  F ) : ran  F -onto-> ( 1st " ran  F )
122 ffn 5389 . . . . . . . . . . . . 13  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  NN )
1232, 122syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  F  Fn  NN )
124 dffn4 5457 . . . . . . . . . . . 12  |-  ( F  Fn  NN  <->  F : NN -onto-> ran  F )
125123, 124sylib 188 . . . . . . . . . . 11  |-  ( ph  ->  F : NN -onto-> ran  F )
126 foco 5461 . . . . . . . . . . 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 ) )
127121, 125, 126sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F ) )
128 fodomnum 7684 . . . . . . . . . 10  |-  ( NN  e.  dom  card  ->  ( ( ( 1st  |`  ran  F
)  o.  F ) : NN -onto-> ( 1st " ran  F )  -> 
( 1st " ran  F )  ~<_  NN ) )
129112, 127, 128mpsyl 59 . . . . . . . . 9  |-  ( ph  ->  ( 1st " ran  F )  ~<_  NN )
130 domentr 6920 . . . . . . . . 9  |-  ( ( ( 1st " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 1st " ran  F )  ~<_  om )
131129, 109, 130sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( 1st " ran  F )  ~<_  om )
132 fofun 5452 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
13381, 132ax-mp 8 . . . . . . . . . . . 12  |-  Fun  2nd
134 fof 5451 . . . . . . . . . . . . . . 15  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
13581, 134ax-mp 8 . . . . . . . . . . . . . 14  |-  2nd : _V
--> _V
136135fdmi 5394 . . . . . . . . . . . . 13  |-  dom  2nd  =  _V
137115, 136sseqtr4i 3211 . . . . . . . . . . . 12  |-  ran  F  C_ 
dom  2nd
138 fores 5460 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  ran  F  C_ 
dom  2nd )  ->  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F ) )
139133, 137, 138mp2an 653 . . . . . . . . . . 11  |-  ( 2nd  |`  ran  F ) : ran  F -onto-> ( 2nd " ran  F )
140 foco 5461 . . . . . . . . . . 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 ) )
141139, 125, 140sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F ) )
142 fodomnum 7684 . . . . . . . . . 10  |-  ( NN  e.  dom  card  ->  ( ( ( 2nd  |`  ran  F
)  o.  F ) : NN -onto-> ( 2nd " ran  F )  -> 
( 2nd " ran  F )  ~<_  NN ) )
143112, 141, 142mpsyl 59 . . . . . . . . 9  |-  ( ph  ->  ( 2nd " ran  F )  ~<_  NN )
144 domentr 6920 . . . . . . . . 9  |-  ( ( ( 2nd " ran  F )  ~<_  NN  /\  NN  ~~  om )  ->  ( 2nd " ran  F )  ~<_  om )
145143, 109, 144sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( 2nd " ran  F )  ~<_  om )
146 unctb 7831 . . . . . . . 8  |-  ( ( ( 1st " ran  F )  ~<_  om  /\  ( 2nd " ran  F )  ~<_  om )  ->  (
( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
147131, 145, 146syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om )
148 reldom 6869 . . . . . . . 8  |-  Rel  ~<_
149148brrelexi 4729 . . . . . . 7  |-  ( ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  ~<_  om  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  e.  _V )
150147, 149syl 15 . . . . . 6  |-  ( ph  ->  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) )  e.  _V )
151 ssid 3197 . . . . . . . 8  |-  U. ran  ( [,]  o.  F ) 
C_  U. ran  ( [,] 
o.  F )
152151, 102syl5sseq 3226 . . . . . . 7  |-  ( ph  ->  U. ran  ( [,] 
o.  F )  C_  ( U. ran  ( (,) 
o.  F )  u.  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) ) )
153 ssundif 3537 . . . . . . 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 ) ) )
154152, 153sylib 188 . . . . . 6  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  C_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
155 ssdomg 6907 . . . . . 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 ) ) ) )
156150, 154, 155sylc 56 . . . . 5  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  ( ( 1st " ran  F )  u.  ( 2nd " ran  F ) ) )
157 domtr 6914 . . . . 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 )
158156, 147, 157syl2anc 642 . . . 4  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om )
159 domentr 6920 . . . 4  |-  ( ( ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  om 
/\  om  ~~  NN )  ->  ( U. ran  ( [,]  o.  F ) 
\  U. ran  ( (,) 
o.  F ) )  ~<_  NN )
160158, 110, 159sylancl 643 . . 3  |-  ( ph  ->  ( U. ran  ( [,]  o.  F )  \  U. ran  ( (,)  o.  F ) )  ~<_  NN )
161 ovolctb2 18851 . . 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 )
162107, 160, 161syl2anc 642 . 2  |-  ( ph  ->  ( vol * `  ( U. ran  ( [,] 
o.  F )  \  U. ran  ( (,)  o.  F ) ) )  =  0 )
163103, 162jca 518 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   {cpr 3641   <.cop 3643   U.cuni 3827   U_ciun 3905   class class class wbr 4023   Oncon0 4392   omcom 4656    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ~~ cen 6860    ~<_ cdom 6861   cardccrd 7568   RRcr 8736   0cc0 8737   RR*cxr 8866    <_ cle 8868   NNcn 9746   (,)cioo 10656   [,]cicc 10659   vol *covol 18822
This theorem is referenced by:  uniioombllem3  18940  uniioombllem4  18941  uniioombllem5  18942  uniiccmbl  18945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824
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