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Theorem dyadmbl 20922
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 20921 . 2  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
5 isfinite 7846 . . . 4  |-  ( G  e.  Fin  <->  G  ~<  om )
6 iccf 11376 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
7 ffun 5549 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
8 funiunfv 5952 . . . . . 6  |-  ( Fun 
[,]  ->  U_ n  e.  G  ( [,] `  n )  =  U. ( [,] " G ) )
96, 7, 8mp2b 10 . . . . 5  |-  U_ n  e.  G  ( [,] `  n )  =  U. ( [,] " G )
10 simpr 458 . . . . . 6  |-  ( (
ph  /\  G  e.  Fin )  ->  G  e. 
Fin )
11 ssrab2 3425 . . . . . . . . . . . . . . . 16  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
122, 11eqsstri 3374 . . . . . . . . . . . . . . 15  |-  G  C_  A
1312, 3syl5ss 3355 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  C_  ran  F )
141dyadf 20913 . . . . . . . . . . . . . . . 16  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
15 frn 5553 . . . . . . . . . . . . . . . 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 3559 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
1816, 17sstri 3353 . . . . . . . . . . . . . 14  |-  ran  F  C_  ( RR  X.  RR )
1913, 18syl6ss 3356 . . . . . . . . . . . . 13  |-  ( ph  ->  G  C_  ( RR  X.  RR ) )
2019adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  e.  Fin )  ->  G  C_  ( RR  X.  RR ) )
2120sselda 3344 . . . . . . . . . . 11  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  n  e.  ( RR  X.  RR ) )
22 1st2nd2 6602 . . . . . . . . . . 11  |-  ( n  e.  ( RR  X.  RR )  ->  n  = 
<. ( 1st `  n
) ,  ( 2nd `  n ) >. )
2321, 22syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >. )
2423fveq2d 5683 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. ) )
25 df-ov 6083 . . . . . . . . 9  |-  ( ( 1st `  n ) [,] ( 2nd `  n
) )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. )
2624, 25syl6eqr 2483 . . . . . . . 8  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( ( 1st `  n
) [,] ( 2nd `  n ) ) )
27 xp1st 6595 . . . . . . . . . 10  |-  ( n  e.  ( RR  X.  RR )  ->  ( 1st `  n )  e.  RR )
2821, 27syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( 1st `  n )  e.  RR )
29 xp2nd 6596 . . . . . . . . . 10  |-  ( n  e.  ( RR  X.  RR )  ->  ( 2nd `  n )  e.  RR )
3021, 29syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( 2nd `  n )  e.  RR )
31 iccmbl 20889 . . . . . . . . 9  |-  ( ( ( 1st `  n
)  e.  RR  /\  ( 2nd `  n )  e.  RR )  -> 
( ( 1st `  n
) [,] ( 2nd `  n ) )  e. 
dom  vol )
3228, 30, 31syl2anc 654 . . . . . . . 8  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  (
( 1st `  n
) [,] ( 2nd `  n ) )  e. 
dom  vol )
3326, 32eqeltrd 2507 . . . . . . 7  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  e. 
dom  vol )
3433ralrimiva 2789 . . . . . 6  |-  ( (
ph  /\  G  e.  Fin )  ->  A. n  e.  G  ( [,] `  n )  e.  dom  vol )
35 finiunmbl 20867 . . . . . 6  |-  ( ( G  e.  Fin  /\  A. n  e.  G  ( [,] `  n )  e.  dom  vol )  ->  U_ n  e.  G  ( [,] `  n )  e.  dom  vol )
3610, 34, 35syl2anc 654 . . . . 5  |-  ( (
ph  /\  G  e.  Fin )  ->  U_ n  e.  G  ( [,] `  n )  e.  dom  vol )
379, 36syl5eqelr 2518 . . . 4  |-  ( (
ph  /\  G  e.  Fin )  ->  U. ( [,] " G )  e. 
dom  vol )
385, 37sylan2br 473 . . 3  |-  ( (
ph  /\  G  ~<  om )  ->  U. ( [,] " G )  e. 
dom  vol )
39 nnenom 11786 . . . . . . 7  |-  NN  ~~  om
40 ensym 7346 . . . . . . 7  |-  ( G 
~~  om  ->  om  ~~  G )
41 entr 7349 . . . . . . 7  |-  ( ( NN  ~~  om  /\  om 
~~  G )  ->  NN  ~~  G )
4239, 40, 41sylancr 656 . . . . . 6  |-  ( G 
~~  om  ->  NN  ~~  G )
43 bren 7307 . . . . . 6  |-  ( NN 
~~  G  <->  E. f 
f : NN -1-1-onto-> G )
4442, 43sylib 196 . . . . 5  |-  ( G 
~~  om  ->  E. f 
f : NN -1-1-onto-> G )
45 rnco2 5333 . . . . . . . . . 10  |-  ran  ( [,]  o.  f )  =  ( [,] " ran  f )
46 f1ofo 5636 . . . . . . . . . . . . 13  |-  ( f : NN -1-1-onto-> G  ->  f : NN -onto-> G )
4746adantl 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN -onto-> G )
48 forn 5611 . . . . . . . . . . . 12  |-  ( f : NN -onto-> G  ->  ran  f  =  G
)
4947, 48syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  f  =  G )
5049imaeq2d 5157 . . . . . . . . . 10  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ( [,] " ran  f )  =  ( [,] " G
) )
5145, 50syl5eq 2477 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  ( [,]  o.  f )  =  ( [,] " G
) )
5251unieqd 4089 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  =  U. ( [,] " G ) )
53 f1of 5629 . . . . . . . . . 10  |-  ( f : NN -1-1-onto-> G  ->  f : NN
--> G )
5413, 16syl6ss 3356 . . . . . . . . . 10  |-  ( ph  ->  G  C_  (  <_  i^i  ( RR  X.  RR ) ) )
55 fss 5555 . . . . . . . . . 10  |-  ( ( f : NN --> G  /\  G  C_  (  <_  i^i  ( RR  X.  RR ) ) )  -> 
f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
5653, 54, 55syl2anr 475 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
57 fss 5555 . . . . . . . . . . . . . . 15  |-  ( ( f : NN --> G  /\  G  C_  ran  F )  ->  f : NN --> ran  F )
5853, 13, 57syl2anr 475 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> ran  F )
59 simpl 454 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  a  e.  NN )
60 ffvelrn 5829 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  F  /\  a  e.  NN )  ->  ( f `  a )  e.  ran  F )
6158, 59, 60syl2an 474 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  ran  F )
62 simpr 458 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  b  e.  NN )
63 ffvelrn 5829 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  F  /\  b  e.  NN )  ->  ( f `  b )  e.  ran  F )
6458, 62, 63syl2an 474 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  ran  F )
651dyaddisj 20918 . . . . . . . . . . . . 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 654 . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> G )
68 ffvelrn 5829 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> G  /\  b  e.  NN )  ->  ( f `  b
)  e.  G )
6967, 62, 68syl2an 474 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  G
)
7012, 69sseldi 3342 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  A
)
71 ffvelrn 5829 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> G  /\  a  e.  NN )  ->  ( f `  a
)  e.  G )
7267, 59, 71syl2an 474 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  G
)
73 fveq2 5679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  a )  ->  ( [,] `  z )  =  ( [,] `  (
f `  a )
) )
7473sseq1d 3371 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  a )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  w ) ) )
75 eqeq1 2439 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  a )  ->  (
z  =  w  <->  ( f `  a )  =  w ) )
7674, 75imbi12d 320 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( f `  a )  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7776ralbidv 2725 . . . . . . . . . . . . . . . . . 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 3108 . . . . . . . . . . . . . . . . 17  |-  ( ( f `  a )  e.  G  <->  ( (
f `  a )  e.  A  /\  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7978simprbi 461 . . . . . . . . . . . . . . . 16  |-  ( ( f `  a )  e.  G  ->  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) )
8072, 79syl 16 . . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  b )  ->  ( [,] `  w )  =  ( [,] `  (
f `  b )
) )
8281sseq2d 3372 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  b )  ->  (
( [,] `  (
f `  a )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  ( f `  b ) ) ) )
83 eqeq2 2442 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  b )  ->  (
( f `  a
)  =  w  <->  ( f `  a )  =  ( f `  b ) ) )
8482, 83imbi12d 320 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( f `  b )  ->  (
( ( [,] `  (
f `  a )
)  C_  ( [,] `  w )  ->  (
f `  a )  =  w )  <->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  ->  ( f `  a )  =  ( f `  b ) ) ) )
8584rspcv 3058 . . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . . 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 5628 . . . . . . . . . . . . . . . . 17  |-  ( f : NN -1-1-onto-> G  ->  f : NN
-1-1-> G )
8887adantl 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN -1-1-> G )
89 f1fveq 5962 . . . . . . . . . . . . . . . 16  |-  ( ( f : NN -1-1-> G  /\  ( a  e.  NN  /\  b  e.  NN ) )  ->  ( (
f `  a )  =  ( f `  b )  <->  a  =  b ) )
9088, 89sylan 468 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
f `  a )  =  ( f `  b )  <->  a  =  b ) )
91 orc 385 . . . . . . . . . . . . . . 15  |-  ( a  =  b  ->  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
9290, 91syl6bi 228 . . . . . . . . . . . . . 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 44 . . . . . . . . . . . . 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 3342 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  A
)
95 fveq2 5679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  b )  ->  ( [,] `  z )  =  ( [,] `  (
f `  b )
) )
9695sseq1d 3371 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  b )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  w ) ) )
97 eqeq1 2439 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  b )  ->  (
z  =  w  <->  ( f `  b )  =  w ) )
9896, 97imbi12d 320 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( f `  b )  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
9998ralbidv 2725 . . . . . . . . . . . . . . . . . 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 3108 . . . . . . . . . . . . . . . . 17  |-  ( ( f `  b )  e.  G  <->  ( (
f `  b )  e.  A  /\  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
101100simprbi 461 . . . . . . . . . . . . . . . 16  |-  ( ( f `  b )  e.  G  ->  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) )
10269, 101syl 16 . . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  ( [,] `  w )  =  ( [,] `  (
f `  a )
) )
104103sseq2d 3372 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  a )  ->  (
( [,] `  (
f `  b )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  ( f `  a ) ) ) )
105 eqeq2 2442 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  (
( f `  b
)  =  w  <->  ( f `  b )  =  ( f `  a ) ) )
106 eqcom 2435 . . . . . . . . . . . . . . . . . 18  |-  ( ( f `  b )  =  ( f `  a )  <->  ( f `  a )  =  ( f `  b ) )
107105, 106syl6bb 261 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  a )  ->  (
( f `  b
)  =  w  <->  ( f `  a )  =  ( f `  b ) ) )
108104, 107imbi12d 320 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( f `  a )  ->  (
( ( [,] `  (
f `  b )
)  C_  ( [,] `  w )  ->  (
f `  b )  =  w )  <->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  ->  ( f `  a )  =  ( f `  b ) ) ) )
109108rspcv 3058 . . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . . 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 44 . . . . . . . . . . . . 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 384 . . . . . . . . . . . . . 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 1275 . . . . . . . . . . . 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 2798 . . . . . . . . . 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 5679 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
f `  a )  =  ( f `  b ) )
118117fveq2d 5683 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( (,) `  ( f `  a ) )  =  ( (,) `  (
f `  b )
) )
119118disjor 4264 . . . . . . . . . 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 212 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  -> Disj  a  e.  NN  ( (,) `  (
f `  a )
) )
121 eqid 2433 . . . . . . . . 9  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
12256, 120, 121uniiccmbl 20912 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  e.  dom  vol )
12352, 122eqeltrrd 2508 . . . . . . 7  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ( [,] " G )  e. 
dom  vol )
124123ex 434 . . . . . 6  |-  ( ph  ->  ( f : NN -1-1-onto-> G  ->  U. ( [,] " G
)  e.  dom  vol ) )
125124exlimdv 1689 . . . . 5  |-  ( ph  ->  ( E. f  f : NN -1-1-onto-> G  ->  U. ( [,] " G )  e. 
dom  vol ) )
12644, 125syl5 32 . . . 4  |-  ( ph  ->  ( G  ~~  om  ->  U. ( [,] " G
)  e.  dom  vol ) )
127126imp 429 . . 3  |-  ( (
ph  /\  G  ~~  om )  ->  U. ( [,] " G )  e. 
dom  vol )
128 reex 9361 . . . . . . . . 9  |-  RR  e.  _V
129128, 128xpex 6497 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
130129inex2 4422 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
131130, 16ssexi 4425 . . . . . 6  |-  ran  F  e.  _V
132 ssdomg 7343 . . . . . 6  |-  ( ran 
F  e.  _V  ->  ( G  C_  ran  F  ->  G  ~<_  ran  F )
)
133131, 13, 132mpsyl 63 . . . . 5  |-  ( ph  ->  G  ~<_  ran  F )
134 omelon 7840 . . . . . . . 8  |-  om  e.  On
135 znnen 13478 . . . . . . . . . . . 12  |-  ZZ  ~~  NN
136135, 39entri 7351 . . . . . . . . . . 11  |-  ZZ  ~~  om
137 nn0ennn 11785 . . . . . . . . . . . 12  |-  NN0  ~~  NN
138137, 39entri 7351 . . . . . . . . . . 11  |-  NN0  ~~  om
139 xpen 7462 . . . . . . . . . . 11  |-  ( ( ZZ  ~~  om  /\  NN0  ~~  om )  ->  ( ZZ  X.  NN0 )  ~~  ( om  X.  om )
)
140136, 138, 139mp2an 665 . . . . . . . . . 10  |-  ( ZZ 
X.  NN0 )  ~~  ( om  X.  om )
141 xpomen 8170 . . . . . . . . . 10  |-  ( om 
X.  om )  ~~  om
142140, 141entri 7351 . . . . . . . . 9  |-  ( ZZ 
X.  NN0 )  ~~  om
143142ensymi 7347 . . . . . . . 8  |-  om  ~~  ( ZZ  X.  NN0 )
144 isnumi 8104 . . . . . . . 8  |-  ( ( om  e.  On  /\  om 
~~  ( ZZ  X.  NN0 ) )  ->  ( ZZ  X.  NN0 )  e. 
dom  card )
145134, 143, 144mp2an 665 . . . . . . 7  |-  ( ZZ 
X.  NN0 )  e.  dom  card
146 ffn 5547 . . . . . . . . 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 5614 . . . . . . . 8  |-  ( F  Fn  ( ZZ  X.  NN0 )  <->  F : ( ZZ 
X.  NN0 ) -onto-> ran  F
)
149147, 148mpbi 208 . . . . . . 7  |-  F :
( ZZ  X.  NN0 ) -onto-> ran  F
150 fodomnum 8215 . . . . . . 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 7356 . . . . . 6  |-  ( ( ran  F  ~<_  ( ZZ 
X.  NN0 )  /\  ( ZZ  X.  NN0 )  ~~  om )  ->  ran  F  ~<_  om )
153151, 142, 152mp2an 665 . . . . 5  |-  ran  F  ~<_  om
154 domtr 7350 . . . . 5  |-  ( ( G  ~<_  ran  F  /\  ran  F  ~<_  om )  ->  G  ~<_  om )
155133, 153, 154sylancl 655 . . . 4  |-  ( ph  ->  G  ~<_  om )
156 brdom2 7327 . . . 4  |-  ( G  ~<_  om  <->  ( G  ~<  om  \/  G  ~~  om ) )
157155, 156sylib 196 . . 3  |-  ( ph  ->  ( G  ~<  om  \/  G  ~~  om ) )
15838, 127, 157mpjaodan 777 . 2  |-  ( ph  ->  U. ( [,] " G
)  e.  dom  vol )
1594, 158eqeltrd 2507 1  |-  ( ph  ->  U. ( [,] " A
)  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 957    = wceq 1362   E.wex 1589    e. wcel 1755   A.wral 2705   {crab 2709   _Vcvv 2962    i^i cin 3315    C_ wss 3316   (/)c0 3625   ~Pcpw 3848   <.cop 3871   U.cuni 4079   U_ciun 4159  Disj wdisj 4250   class class class wbr 4280   Oncon0 4706    X. cxp 4825   dom cdm 4827   ran crn 4828   "cima 4830    o. ccom 4831   Fun wfun 5400    Fn wfn 5401   -->wf 5402   -1-1->wf1 5403   -onto->wfo 5404   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   omcom 6465   1stc1st 6564   2ndc2nd 6565    ~~ cen 7295    ~<_ cdom 7296    ~< csdm 7297   Fincfn 7298   cardccrd 8093   RRcr 9269   1c1 9271    + caddc 9273   RR*cxr 9405    <_ cle 9407    - cmin 9583    / cdiv 9981   NNcn 10310   2c2 10359   NN0cn0 10567   ZZcz 10634   (,)cioo 11288   [,]cicc 11291    seqcseq 11790   ^cexp 11849   abscabs 12707   volcvol 20789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-omul 6913  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-acn 8100  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-rlim 12951  df-sum 13148  df-rest 14344  df-topgen 14365  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-top 18345  df-bases 18347  df-topon 18348  df-cmp 18832  df-ovol 20790  df-vol 20791
This theorem is referenced by:  opnmbllem  20923
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