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Theorem dyadmbl 22607
Description: Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypotheses
Ref Expression
dyadmbl.1  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
dyadmbl.2  |-  G  =  { z  e.  A  |  A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }
dyadmbl.3  |-  ( ph  ->  A  C_  ran  F )
Assertion
Ref Expression
dyadmbl  |-  ( ph  ->  U. ( [,] " A
)  e.  dom  vol )
Distinct variable groups:    x, y    z, w, ph    x, w, y, A, z    z, G   
w, F, x, y, z
Allowed substitution hints:    ph( x, y)    G( x, y, w)

Proof of Theorem dyadmbl
Dummy variables  f 
a  b  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dyadmbl.1 . . 3  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
2 dyadmbl.2 . . 3  |-  G  =  { z  e.  A  |  A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }
3 dyadmbl.3 . . 3  |-  ( ph  ->  A  C_  ran  F )
41, 2, 3dyadmbllem 22606 . 2  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
5 isfinite 8183 . . . 4  |-  ( G  e.  Fin  <->  G  ~<  om )
6 iccf 11762 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
7 ffun 5754 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
8 funiunfv 6178 . . . . . 6  |-  ( Fun 
[,]  ->  U_ n  e.  G  ( [,] `  n )  =  U. ( [,] " G ) )
96, 7, 8mp2b 10 . . . . 5  |-  U_ n  e.  G  ( [,] `  n )  =  U. ( [,] " G )
10 simpr 467 . . . . . 6  |-  ( (
ph  /\  G  e.  Fin )  ->  G  e. 
Fin )
11 ssrab2 3526 . . . . . . . . . . . . . . . 16  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
122, 11eqsstri 3474 . . . . . . . . . . . . . . 15  |-  G  C_  A
1312, 3syl5ss 3455 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  C_  ran  F )
141dyadf 22598 . . . . . . . . . . . . . . . 16  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
15 frn 5758 . . . . . . . . . . . . . . . 16  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
1614, 15ax-mp 5 . . . . . . . . . . . . . . 15  |-  ran  F  C_  (  <_  i^i  ( RR  X.  RR ) )
17 inss2 3665 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
1816, 17sstri 3453 . . . . . . . . . . . . . 14  |-  ran  F  C_  ( RR  X.  RR )
1913, 18syl6ss 3456 . . . . . . . . . . . . 13  |-  ( ph  ->  G  C_  ( RR  X.  RR ) )
2019adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  e.  Fin )  ->  G  C_  ( RR  X.  RR ) )
2120sselda 3444 . . . . . . . . . . 11  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  n  e.  ( RR  X.  RR ) )
22 1st2nd2 6857 . . . . . . . . . . 11  |-  ( n  e.  ( RR  X.  RR )  ->  n  = 
<. ( 1st `  n
) ,  ( 2nd `  n ) >. )
2321, 22syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >. )
2423fveq2d 5892 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. ) )
25 df-ov 6318 . . . . . . . . 9  |-  ( ( 1st `  n ) [,] ( 2nd `  n
) )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. )
2624, 25syl6eqr 2514 . . . . . . . 8  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( ( 1st `  n
) [,] ( 2nd `  n ) ) )
27 xp1st 6850 . . . . . . . . . 10  |-  ( n  e.  ( RR  X.  RR )  ->  ( 1st `  n )  e.  RR )
2821, 27syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( 1st `  n )  e.  RR )
29 xp2nd 6851 . . . . . . . . . 10  |-  ( n  e.  ( RR  X.  RR )  ->  ( 2nd `  n )  e.  RR )
3021, 29syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( 2nd `  n )  e.  RR )
31 iccmbl 22568 . . . . . . . . 9  |-  ( ( ( 1st `  n
)  e.  RR  /\  ( 2nd `  n )  e.  RR )  -> 
( ( 1st `  n
) [,] ( 2nd `  n ) )  e. 
dom  vol )
3228, 30, 31syl2anc 671 . . . . . . . 8  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  (
( 1st `  n
) [,] ( 2nd `  n ) )  e. 
dom  vol )
3326, 32eqeltrd 2540 . . . . . . 7  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  e. 
dom  vol )
3433ralrimiva 2814 . . . . . 6  |-  ( (
ph  /\  G  e.  Fin )  ->  A. n  e.  G  ( [,] `  n )  e.  dom  vol )
35 finiunmbl 22546 . . . . . 6  |-  ( ( G  e.  Fin  /\  A. n  e.  G  ( [,] `  n )  e.  dom  vol )  ->  U_ n  e.  G  ( [,] `  n )  e.  dom  vol )
3610, 34, 35syl2anc 671 . . . . 5  |-  ( (
ph  /\  G  e.  Fin )  ->  U_ n  e.  G  ( [,] `  n )  e.  dom  vol )
379, 36syl5eqelr 2545 . . . 4  |-  ( (
ph  /\  G  e.  Fin )  ->  U. ( [,] " G )  e. 
dom  vol )
385, 37sylan2br 483 . . 3  |-  ( (
ph  /\  G  ~<  om )  ->  U. ( [,] " G )  e. 
dom  vol )
39 nnenom 12225 . . . . . . 7  |-  NN  ~~  om
40 ensym 7644 . . . . . . 7  |-  ( G 
~~  om  ->  om  ~~  G )
41 entr 7647 . . . . . . 7  |-  ( ( NN  ~~  om  /\  om 
~~  G )  ->  NN  ~~  G )
4239, 40, 41sylancr 674 . . . . . 6  |-  ( G 
~~  om  ->  NN  ~~  G )
43 bren 7604 . . . . . 6  |-  ( NN 
~~  G  <->  E. f 
f : NN -1-1-onto-> G )
4442, 43sylib 201 . . . . 5  |-  ( G 
~~  om  ->  E. f 
f : NN -1-1-onto-> G )
45 rnco2 5361 . . . . . . . . . 10  |-  ran  ( [,]  o.  f )  =  ( [,] " ran  f )
46 f1ofo 5844 . . . . . . . . . . . . 13  |-  ( f : NN -1-1-onto-> G  ->  f : NN -onto-> G )
4746adantl 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN -onto-> G )
48 forn 5819 . . . . . . . . . . . 12  |-  ( f : NN -onto-> G  ->  ran  f  =  G
)
4947, 48syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  f  =  G )
5049imaeq2d 5187 . . . . . . . . . 10  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ( [,] " ran  f )  =  ( [,] " G
) )
5145, 50syl5eq 2508 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  ( [,]  o.  f )  =  ( [,] " G
) )
5251unieqd 4222 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  =  U. ( [,] " G ) )
53 f1of 5837 . . . . . . . . . 10  |-  ( f : NN -1-1-onto-> G  ->  f : NN
--> G )
5413, 16syl6ss 3456 . . . . . . . . . 10  |-  ( ph  ->  G  C_  (  <_  i^i  ( RR  X.  RR ) ) )
55 fss 5760 . . . . . . . . . 10  |-  ( ( f : NN --> G  /\  G  C_  (  <_  i^i  ( RR  X.  RR ) ) )  -> 
f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
5653, 54, 55syl2anr 485 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
57 fss 5760 . . . . . . . . . . . . . . 15  |-  ( ( f : NN --> G  /\  G  C_  ran  F )  ->  f : NN --> ran  F )
5853, 13, 57syl2anr 485 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> ran  F )
59 simpl 463 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  a  e.  NN )
60 ffvelrn 6043 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  F  /\  a  e.  NN )  ->  ( f `  a )  e.  ran  F )
6158, 59, 60syl2an 484 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  ran  F )
62 simpr 467 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  b  e.  NN )
63 ffvelrn 6043 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  F  /\  b  e.  NN )  ->  ( f `  b )  e.  ran  F )
6458, 62, 63syl2an 484 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  ran  F )
651dyaddisj 22603 . . . . . . . . . . . . 13  |-  ( ( ( f `  a
)  e.  ran  F  /\  ( f `  b
)  e.  ran  F
)  ->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  \/  ( [,] `  (
f `  b )
)  C_  ( [,] `  ( f `  a
) )  \/  (
( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
6661, 64, 65syl2anc 671 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  \/  ( [,] `  (
f `  b )
)  C_  ( [,] `  ( f `  a
) )  \/  (
( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
6753adantl 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> G )
68 ffvelrn 6043 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> G  /\  b  e.  NN )  ->  ( f `  b
)  e.  G )
6967, 62, 68syl2an 484 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  G
)
7012, 69sseldi 3442 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  A
)
71 ffvelrn 6043 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> G  /\  a  e.  NN )  ->  ( f `  a
)  e.  G )
7267, 59, 71syl2an 484 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  G
)
73 fveq2 5888 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  a )  ->  ( [,] `  z )  =  ( [,] `  (
f `  a )
) )
7473sseq1d 3471 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  a )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  w ) ) )
75 eqeq1 2466 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  a )  ->  (
z  =  w  <->  ( f `  a )  =  w ) )
7674, 75imbi12d 326 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( f `  a )  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7776ralbidv 2839 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  ( f `  a )  ->  ( A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7877, 2elrab2 3210 . . . . . . . . . . . . . . . . 17  |-  ( ( f `  a )  e.  G  <->  ( (
f `  a )  e.  A  /\  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7978simprbi 470 . . . . . . . . . . . . . . . 16  |-  ( ( f `  a )  e.  G  ->  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) )
8072, 79syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) )
81 fveq2 5888 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  b )  ->  ( [,] `  w )  =  ( [,] `  (
f `  b )
) )
8281sseq2d 3472 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  b )  ->  (
( [,] `  (
f `  a )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  ( f `  b ) ) ) )
83 eqeq2 2473 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  b )  ->  (
( f `  a
)  =  w  <->  ( f `  a )  =  ( f `  b ) ) )
8482, 83imbi12d 326 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( f `  b )  ->  (
( ( [,] `  (
f `  a )
)  C_  ( [,] `  w )  ->  (
f `  a )  =  w )  <->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  ->  ( f `  a )  =  ( f `  b ) ) ) )
8584rspcv 3158 . . . . . . . . . . . . . . 15  |-  ( ( f `  b )  e.  A  ->  ( A. w  e.  A  ( ( [,] `  (
f `  a )
)  C_  ( [,] `  w )  ->  (
f `  a )  =  w )  ->  (
( [,] `  (
f `  a )
)  C_  ( [,] `  ( f `  b
) )  ->  (
f `  a )  =  ( f `  b ) ) ) )
8670, 80, 85sylc 62 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  ->  ( f `  a )  =  ( f `  b ) ) )
87 f1of1 5836 . . . . . . . . . . . . . . . . 17  |-  ( f : NN -1-1-onto-> G  ->  f : NN
-1-1-> G )
8887adantl 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN -1-1-> G )
89 f1fveq 6188 . . . . . . . . . . . . . . . 16  |-  ( ( f : NN -1-1-> G  /\  ( a  e.  NN  /\  b  e.  NN ) )  ->  ( (
f `  a )  =  ( f `  b )  <->  a  =  b ) )
9088, 89sylan 478 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
f `  a )  =  ( f `  b )  <->  a  =  b ) )
91 orc 391 . . . . . . . . . . . . . . 15  |-  ( a  =  b  ->  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
9290, 91syl6bi 236 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
f `  a )  =  ( f `  b )  ->  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) ) )
9386, 92syld 45 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  ->  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) ) )
9412, 72sseldi 3442 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  A
)
95 fveq2 5888 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  b )  ->  ( [,] `  z )  =  ( [,] `  (
f `  b )
) )
9695sseq1d 3471 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  b )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  w ) ) )
97 eqeq1 2466 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  b )  ->  (
z  =  w  <->  ( f `  b )  =  w ) )
9896, 97imbi12d 326 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( f `  b )  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
9998ralbidv 2839 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  ( f `  b )  ->  ( A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
10099, 2elrab2 3210 . . . . . . . . . . . . . . . . 17  |-  ( ( f `  b )  e.  G  <->  ( (
f `  b )  e.  A  /\  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
101100simprbi 470 . . . . . . . . . . . . . . . 16  |-  ( ( f `  b )  e.  G  ->  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) )
10269, 101syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) )
103 fveq2 5888 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  ( [,] `  w )  =  ( [,] `  (
f `  a )
) )
104103sseq2d 3472 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  a )  ->  (
( [,] `  (
f `  b )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  ( f `  a ) ) ) )
105 eqeq2 2473 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  (
( f `  b
)  =  w  <->  ( f `  b )  =  ( f `  a ) ) )
106 eqcom 2469 . . . . . . . . . . . . . . . . . 18  |-  ( ( f `  b )  =  ( f `  a )  <->  ( f `  a )  =  ( f `  b ) )
107105, 106syl6bb 269 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  a )  ->  (
( f `  b
)  =  w  <->  ( f `  a )  =  ( f `  b ) ) )
108104, 107imbi12d 326 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( f `  a )  ->  (
( ( [,] `  (
f `  b )
)  C_  ( [,] `  w )  ->  (
f `  b )  =  w )  <->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  ->  ( f `  a )  =  ( f `  b ) ) ) )
109108rspcv 3158 . . . . . . . . . . . . . . 15  |-  ( ( f `  a )  e.  A  ->  ( A. w  e.  A  ( ( [,] `  (
f `  b )
)  C_  ( [,] `  w )  ->  (
f `  b )  =  w )  ->  (
( [,] `  (
f `  b )
)  C_  ( [,] `  ( f `  a
) )  ->  (
f `  a )  =  ( f `  b ) ) ) )
11094, 102, 109sylc 62 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  ->  ( f `  a )  =  ( f `  b ) ) )
111110, 92syld 45 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  ->  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) ) )
112 olc 390 . . . . . . . . . . . . . 14  |-  ( ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/)  ->  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) )
113112a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/)  ->  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) ) )
11493, 111, 1133jaod 1341 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
( [,] `  (
f `  a )
)  C_  ( [,] `  ( f `  b
) )  \/  ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  \/  ( ( (,) `  ( f `  a
) )  i^i  ( (,) `  ( f `  b ) ) )  =  (/) )  ->  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) ) )
11566, 114mpd 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( a  =  b  \/  (
( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
116115ralrimivva 2821 . . . . . . . . . 10  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  A. a  e.  NN  A. b  e.  NN  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) )
117 fveq2 5888 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
f `  a )  =  ( f `  b ) )
118117fveq2d 5892 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( (,) `  ( f `  a ) )  =  ( (,) `  (
f `  b )
) )
119118disjor 4401 . . . . . . . . . 10  |-  (Disj  a  e.  NN  ( (,) `  (
f `  a )
)  <->  A. a  e.  NN  A. b  e.  NN  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
120116, 119sylibr 217 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  -> Disj  a  e.  NN  ( (,) `  (
f `  a )
) )
121 eqid 2462 . . . . . . . . 9  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
12256, 120, 121uniiccmbl 22597 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  e.  dom  vol )
12352, 122eqeltrrd 2541 . . . . . . 7  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ( [,] " G )  e. 
dom  vol )
124123ex 440 . . . . . 6  |-  ( ph  ->  ( f : NN -1-1-onto-> G  ->  U. ( [,] " G
)  e.  dom  vol ) )
125124exlimdv 1790 . . . . 5  |-  ( ph  ->  ( E. f  f : NN -1-1-onto-> G  ->  U. ( [,] " G )  e. 
dom  vol ) )
12644, 125syl5 33 . . . 4  |-  ( ph  ->  ( G  ~~  om  ->  U. ( [,] " G
)  e.  dom  vol ) )
127126imp 435 . . 3  |-  ( (
ph  /\  G  ~~  om )  ->  U. ( [,] " G )  e. 
dom  vol )
128 reex 9656 . . . . . . . . 9  |-  RR  e.  _V
129128, 128xpex 6622 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
130129inex2 4559 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
131130, 16ssexi 4562 . . . . . 6  |-  ran  F  e.  _V
132 ssdomg 7641 . . . . . 6  |-  ( ran 
F  e.  _V  ->  ( G  C_  ran  F  ->  G  ~<_  ran  F )
)
133131, 13, 132mpsyl 65 . . . . 5  |-  ( ph  ->  G  ~<_  ran  F )
134 omelon 8177 . . . . . . . 8  |-  om  e.  On
135 znnen 14314 . . . . . . . . . . . 12  |-  ZZ  ~~  NN
136135, 39entri 7649 . . . . . . . . . . 11  |-  ZZ  ~~  om
137 nn0ennn 12224 . . . . . . . . . . . 12  |-  NN0  ~~  NN
138137, 39entri 7649 . . . . . . . . . . 11  |-  NN0  ~~  om
139 xpen 7761 . . . . . . . . . . 11  |-  ( ( ZZ  ~~  om  /\  NN0  ~~  om )  ->  ( ZZ  X.  NN0 )  ~~  ( om  X.  om )
)
140136, 138, 139mp2an 683 . . . . . . . . . 10  |-  ( ZZ 
X.  NN0 )  ~~  ( om  X.  om )
141 xpomen 8472 . . . . . . . . . 10  |-  ( om 
X.  om )  ~~  om
142140, 141entri 7649 . . . . . . . . 9  |-  ( ZZ 
X.  NN0 )  ~~  om
143142ensymi 7645 . . . . . . . 8  |-  om  ~~  ( ZZ  X.  NN0 )
144 isnumi 8406 . . . . . . . 8  |-  ( ( om  e.  On  /\  om 
~~  ( ZZ  X.  NN0 ) )  ->  ( ZZ  X.  NN0 )  e. 
dom  card )
145134, 143, 144mp2an 683 . . . . . . 7  |-  ( ZZ 
X.  NN0 )  e.  dom  card
146 ffn 5751 . . . . . . . . 9  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
14714, 146ax-mp 5 . . . . . . . 8  |-  F  Fn  ( ZZ  X.  NN0 )
148 dffn4 5822 . . . . . . . 8  |-  ( F  Fn  ( ZZ  X.  NN0 )  <->  F : ( ZZ 
X.  NN0 ) -onto-> ran  F
)
149147, 148mpbi 213 . . . . . . 7  |-  F :
( ZZ  X.  NN0 ) -onto-> ran  F
150 fodomnum 8514 . . . . . . 7  |-  ( ( ZZ  X.  NN0 )  e.  dom  card  ->  ( F : ( ZZ  X.  NN0 ) -onto-> ran  F  ->  ran  F  ~<_  ( ZZ  X.  NN0 ) ) )
151145, 149, 150mp2 9 . . . . . 6  |-  ran  F  ~<_  ( ZZ  X.  NN0 )
152 domentr 7654 . . . . . 6  |-  ( ( ran  F  ~<_  ( ZZ 
X.  NN0 )  /\  ( ZZ  X.  NN0 )  ~~  om )  ->  ran  F  ~<_  om )
153151, 142, 152mp2an 683 . . . . 5  |-  ran  F  ~<_  om
154 domtr 7648 . . . . 5  |-  ( ( G  ~<_  ran  F  /\  ran  F  ~<_  om )  ->  G  ~<_  om )
155133, 153, 154sylancl 673 . . . 4  |-  ( ph  ->  G  ~<_  om )
156 brdom2 7625 . . . 4  |-  ( G  ~<_  om  <->  ( G  ~<  om  \/  G  ~~  om ) )
157155, 156sylib 201 . . 3  |-  ( ph  ->  ( G  ~<  om  \/  G  ~~  om ) )
15838, 127, 157mpjaodan 800 . 2  |-  ( ph  ->  U. ( [,] " G
)  e.  dom  vol )
1594, 158eqeltrd 2540 1  |-  ( ph  ->  U. ( [,] " A
)  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    \/ w3o 990    = wceq 1455   E.wex 1674    e. wcel 1898   A.wral 2749   {crab 2753   _Vcvv 3057    i^i cin 3415    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   <.cop 3986   U.cuni 4212   U_ciun 4292  Disj wdisj 4387   class class class wbr 4416    X. cxp 4851   dom cdm 4853   ran crn 4854   "cima 4856    o. ccom 4857   Oncon0 5442   Fun wfun 5595    Fn wfn 5596   -->wf 5597   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6315    |-> cmpt2 6317   omcom 6719   1stc1st 6818   2ndc2nd 6819    ~~ cen 7592    ~<_ cdom 7593    ~< csdm 7594   Fincfn 7595   cardccrd 8395   RRcr 9564   1c1 9566    + caddc 9568   RR*cxr 9700    <_ cle 9702    - cmin 9886    / cdiv 10297   NNcn 10637   2c2 10687   NN0cn0 10898   ZZcz 10966   (,)cioo 11664   [,]cicc 11667    seqcseq 12245   ^cexp 12304   abscabs 13346   volcvol 22464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-disj 4388  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-omul 7213  df-er 7389  df-map 7500  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fi 7951  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-acn 8402  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-n0 10899  df-z 10967  df-uz 11189  df-q 11294  df-rp 11332  df-xneg 11438  df-xadd 11439  df-xmul 11440  df-ioo 11668  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-fl 12060  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-clim 13601  df-rlim 13602  df-sum 13802  df-rest 15370  df-topgen 15391  df-psmet 19011  df-xmet 19012  df-met 19013  df-bl 19014  df-mopn 19015  df-top 19970  df-bases 19971  df-topon 19972  df-cmp 20451  df-ovol 22465  df-vol 22467
This theorem is referenced by:  opnmbllem  22608
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