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Theorem dyadmbllem 22606
Description: Lemma for dyadmbl 22607. (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
dyadmbllem  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
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 dyadmbllem
Dummy variables  a  m  t  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 4216 . . . 4  |-  ( a  e.  U. ( [,] " A )  <->  E. i  e.  ( [,] " A
) a  e.  i )
2 iccf 11762 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5751 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
42, 3ax-mp 5 . . . . . 6  |-  [,]  Fn  ( RR*  X.  RR* )
5 dyadmbl.3 . . . . . . 7  |-  ( ph  ->  A  C_  ran  F )
6 dyadmbl.1 . . . . . . . . . 10  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
76dyadf 22598 . . . . . . . . 9  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
8 frn 5758 . . . . . . . . 9  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
97, 8ax-mp 5 . . . . . . . 8  |-  ran  F  C_  (  <_  i^i  ( RR  X.  RR ) )
10 inss2 3665 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
11 rexpssxrxp 9711 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
1210, 11sstri 3453 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
139, 12sstri 3453 . . . . . . 7  |-  ran  F  C_  ( RR*  X.  RR* )
145, 13syl6ss 3456 . . . . . 6  |-  ( ph  ->  A  C_  ( RR*  X. 
RR* ) )
15 eleq2 2529 . . . . . . 7  |-  ( i  =  ( [,] `  t
)  ->  ( a  e.  i  <->  a  e.  ( [,] `  t ) ) )
1615rexima 6169 . . . . . 6  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  A  C_  ( RR*  X.  RR* )
)  ->  ( E. i  e.  ( [,] " A ) a  e.  i  <->  E. t  e.  A  a  e.  ( [,] `  t ) ) )
174, 14, 16sylancr 674 . . . . 5  |-  ( ph  ->  ( E. i  e.  ( [,] " A
) a  e.  i  <->  E. t  e.  A  a  e.  ( [,] `  t ) ) )
18 ssrab2 3526 . . . . . . . . 9  |-  { a  e.  A  |  ( [,] `  t ) 
C_  ( [,] `  a
) }  C_  A
195adantr 471 . . . . . . . . 9  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  A  C_  ran  F )
2018, 19syl5ss 3455 . . . . . . . 8  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  C_  ran  F )
21 simprl 769 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  -> 
t  e.  A )
22 ssid 3463 . . . . . . . . . 10  |-  ( [,] `  t )  C_  ( [,] `  t )
23 fveq2 5888 . . . . . . . . . . . 12  |-  ( a  =  t  ->  ( [,] `  a )  =  ( [,] `  t
) )
2423sseq2d 3472 . . . . . . . . . . 11  |-  ( a  =  t  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  t ) ) )
2524rspcev 3162 . . . . . . . . . 10  |-  ( ( t  e.  A  /\  ( [,] `  t ) 
C_  ( [,] `  t
) )  ->  E. a  e.  A  ( [,] `  t )  C_  ( [,] `  a ) )
2621, 22, 25sylancl 673 . . . . . . . . 9  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  E. a  e.  A  ( [,] `  t ) 
C_  ( [,] `  a
) )
27 rabn0 3764 . . . . . . . . 9  |-  ( { a  e.  A  | 
( [,] `  t
)  C_  ( [,] `  a ) }  =/=  (/)  <->  E. a  e.  A  ( [,] `  t ) 
C_  ( [,] `  a
) )
2826, 27sylibr 217 . . . . . . . 8  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  =/=  (/) )
296dyadmax 22605 . . . . . . . 8  |-  ( ( { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  C_  ran  F  /\  { a  e.  A  |  ( [,] `  t ) 
C_  ( [,] `  a
) }  =/=  (/) )  ->  E. m  e.  { a  e.  A  |  ( [,] `  t ) 
C_  ( [,] `  a
) } A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) )
3020, 28, 29syl2anc 671 . . . . . . 7  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  E. m  e.  { a  e.  A  |  ( [,] `  t ) 
C_  ( [,] `  a
) } A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) )
31 fveq2 5888 . . . . . . . . . . 11  |-  ( a  =  m  ->  ( [,] `  a )  =  ( [,] `  m
) )
3231sseq2d 3472 . . . . . . . . . 10  |-  ( a  =  m  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  m ) ) )
3332elrab 3208 . . . . . . . . 9  |-  ( m  e.  { a  e.  A  |  ( [,] `  t )  C_  ( [,] `  a ) }  <-> 
( m  e.  A  /\  ( [,] `  t
)  C_  ( [,] `  m ) ) )
34 simprlr 778 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  ( [,] `  t
)  C_  ( [,] `  m ) )
35 simplrr 776 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  a  e.  ( [,] `  t ) )
3634, 35sseldd 3445 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  a  e.  ( [,] `  m ) )
37 simprll 777 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  m  e.  A
)
38 fveq2 5888 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  w  ->  ( [,] `  a )  =  ( [,] `  w
) )
3938sseq2d 3472 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  w  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  w ) ) )
4039elrab 3208 . . . . . . . . . . . . . . . . . 18  |-  ( w  e.  { a  e.  A  |  ( [,] `  t )  C_  ( [,] `  a ) }  <-> 
( w  e.  A  /\  ( [,] `  t
)  C_  ( [,] `  w ) ) )
4140imbi1i 331 . . . . . . . . . . . . . . . . 17  |-  ( ( w  e.  { a  e.  A  |  ( [,] `  t ) 
C_  ( [,] `  a
) }  ->  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) )  <->  ( (
w  e.  A  /\  ( [,] `  t ) 
C_  ( [,] `  w
) )  ->  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )
42 impexp 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  e.  A  /\  ( [,] `  t
)  C_  ( [,] `  w ) )  -> 
( ( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) )  <->  ( w  e.  A  ->  ( ( [,] `  t ) 
C_  ( [,] `  w
)  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) ) )
4341, 42bitri 257 . . . . . . . . . . . . . . . 16  |-  ( ( w  e.  { a  e.  A  |  ( [,] `  t ) 
C_  ( [,] `  a
) }  ->  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) )  <->  ( w  e.  A  ->  ( ( [,] `  t ) 
C_  ( [,] `  w
)  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) ) )
44 impexp 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( [,] `  t
)  C_  ( [,] `  w )  /\  ( [,] `  m )  C_  ( [,] `  w ) )  ->  m  =  w )  <->  ( ( [,] `  t )  C_  ( [,] `  w )  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w )
) )
45 sstr2 3451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( [,] `  t ) 
C_  ( [,] `  m
)  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  ( [,] `  t
)  C_  ( [,] `  w ) ) )
4645ad2antll 740 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  ( [,] `  t
)  C_  ( [,] `  w ) ) )
4746ancrd 561 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  ( ( [,] `  t )  C_  ( [,] `  w )  /\  ( [,] `  m ) 
C_  ( [,] `  w
) ) ) )
4847imim1d 78 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( (
( ( [,] `  t
)  C_  ( [,] `  w )  /\  ( [,] `  m )  C_  ( [,] `  w ) )  ->  m  =  w )  ->  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )
4944, 48syl5bir 226 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( (
( [,] `  t
)  C_  ( [,] `  w )  ->  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) )  -> 
( ( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )
5049imim2d 54 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( (
w  e.  A  -> 
( ( [,] `  t
)  C_  ( [,] `  w )  ->  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  ( w  e.  A  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) ) )
5143, 50syl5bi 225 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( (
w  e.  { a  e.  A  |  ( [,] `  t ) 
C_  ( [,] `  a
) }  ->  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) )  -> 
( w  e.  A  ->  ( ( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) ) )
5251ralimdv2 2807 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( A. w  e.  { a  e.  A  |  ( [,] `  t )  C_  ( [,] `  a ) }  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w )  ->  A. w  e.  A  ( ( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )
5352impr 629 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  A. w  e.  A  ( ( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) )
54 fveq2 5888 . . . . . . . . . . . . . . . . 17  |-  ( z  =  m  ->  ( [,] `  z )  =  ( [,] `  m
) )
5554sseq1d 3471 . . . . . . . . . . . . . . . 16  |-  ( z  =  m  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  m )  C_  ( [,] `  w ) ) )
56 equequ1 1878 . . . . . . . . . . . . . . . 16  |-  ( z  =  m  ->  (
z  =  w  <->  m  =  w ) )
5755, 56imbi12d 326 . . . . . . . . . . . . . . 15  |-  ( z  =  m  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
5857ralbidv 2839 . . . . . . . . . . . . . 14  |-  ( z  =  m  ->  ( A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  A. w  e.  A  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
59 dyadmbl.2 . . . . . . . . . . . . . 14  |-  G  =  { z  e.  A  |  A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }
6058, 59elrab2 3210 . . . . . . . . . . . . 13  |-  ( m  e.  G  <->  ( m  e.  A  /\  A. w  e.  A  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
6137, 53, 60sylanbrc 675 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  m  e.  G
)
62 ffun 5754 . . . . . . . . . . . . . 14  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
632, 62ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  [,]
64 ssrab2 3526 . . . . . . . . . . . . . . . . 17  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
6559, 64eqsstri 3474 . . . . . . . . . . . . . . . 16  |-  G  C_  A
6665, 14syl5ss 3455 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  C_  ( RR*  X. 
RR* ) )
672fdmi 5757 . . . . . . . . . . . . . . 15  |-  dom  [,]  =  ( RR*  X.  RR* )
6866, 67syl6sseqr 3491 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  C_  dom  [,] )
6968ad2antrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  G  C_  dom  [,] )
70 funfvima2 6166 . . . . . . . . . . . . 13  |-  ( ( Fun  [,]  /\  G  C_  dom  [,] )  ->  (
m  e.  G  -> 
( [,] `  m
)  e.  ( [,] " G ) ) )
7163, 69, 70sylancr 674 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  ( m  e.  G  ->  ( [,] `  m )  e.  ( [,] " G ) ) )
7261, 71mpd 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  ( [,] `  m
)  e.  ( [,] " G ) )
73 elunii 4217 . . . . . . . . . . 11  |-  ( ( a  e.  ( [,] `  m )  /\  ( [,] `  m )  e.  ( [,] " G
) )  ->  a  e.  U. ( [,] " G
) )
7436, 72, 73syl2anc 671 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  /\  A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) ) )  ->  a  e.  U. ( [,] " G ) )
7574exp32 614 . . . . . . . . 9  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  -> 
( ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) )  ->  ( A. w  e.  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w )  ->  a  e.  U. ( [,] " G
) ) ) )
7633, 75syl5bi 225 . . . . . . . 8  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  -> 
( m  e.  {
a  e.  A  | 
( [,] `  t
)  C_  ( [,] `  a ) }  ->  ( A. w  e.  {
a  e.  A  | 
( [,] `  t
)  C_  ( [,] `  a ) }  (
( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w )  ->  a  e.  U. ( [,] " G
) ) ) )
7776rexlimdv 2889 . . . . . . 7  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  -> 
( E. m  e. 
{ a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) } A. w  e.  { a  e.  A  |  ( [,] `  t )  C_  ( [,] `  a ) }  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w )  ->  a  e.  U. ( [,] " G ) ) )
7830, 77mpd 15 . . . . . 6  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  -> 
a  e.  U. ( [,] " G ) )
7978rexlimdvaa 2892 . . . . 5  |-  ( ph  ->  ( E. t  e.  A  a  e.  ( [,] `  t )  ->  a  e.  U. ( [,] " G ) ) )
8017, 79sylbid 223 . . . 4  |-  ( ph  ->  ( E. i  e.  ( [,] " A
) a  e.  i  ->  a  e.  U. ( [,] " G ) ) )
811, 80syl5bi 225 . . 3  |-  ( ph  ->  ( a  e.  U. ( [,] " A )  ->  a  e.  U. ( [,] " G ) ) )
8281ssrdv 3450 . 2  |-  ( ph  ->  U. ( [,] " A
)  C_  U. ( [,] " G ) )
83 imass2 5223 . . . 4  |-  ( G 
C_  A  ->  ( [,] " G )  C_  ( [,] " A ) )
8465, 83ax-mp 5 . . 3  |-  ( [,] " G )  C_  ( [,] " A )
85 uniss 4233 . . 3  |-  ( ( [,] " G ) 
C_  ( [,] " A
)  ->  U. ( [,] " G )  C_  U. ( [,] " A
) )
8684, 85mp1i 13 . 2  |-  ( ph  ->  U. ( [,] " G
)  C_  U. ( [,] " A ) )
8782, 86eqssd 3461 1  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753    i^i cin 3415    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   <.cop 3986   U.cuni 4212    X. cxp 4851   dom cdm 4853   ran crn 4854   "cima 4856   Fun wfun 5595    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6315    |-> cmpt2 6317   RRcr 9564   1c1 9566    + caddc 9568   RR*cxr 9700    <_ cle 9702    / cdiv 10297   2c2 10687   NN0cn0 10898   ZZcz 10966   [,]cicc 11667   ^cexp 12304
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-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-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  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-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-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-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-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
This theorem is referenced by:  dyadmbl  22607  mblfinlem1  32022  mblfinlem2  32023
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