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Theorem dyadmbllem 21874
Description: Lemma for dyadmbl 21875. (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 4234 . . . 4  |-  ( a  e.  U. ( [,] " A )  <->  E. i  e.  ( [,] " A
) a  e.  i )
2 iccf 11627 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5717 . . . . . . 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 21866 . . . . . . . . 9  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
8 frn 5723 . . . . . . . . 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 3701 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
11 rexpssxrxp 9636 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
1210, 11sstri 3495 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
139, 12sstri 3495 . . . . . . 7  |-  ran  F  C_  ( RR*  X.  RR* )
145, 13syl6ss 3498 . . . . . 6  |-  ( ph  ->  A  C_  ( RR*  X. 
RR* ) )
15 eleq2 2514 . . . . . . 7  |-  ( i  =  ( [,] `  t
)  ->  ( a  e.  i  <->  a  e.  ( [,] `  t ) ) )
1615rexima 6132 . . . . . 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 663 . . . . 5  |-  ( ph  ->  ( E. i  e.  ( [,] " A
) a  e.  i  <->  E. t  e.  A  a  e.  ( [,] `  t ) ) )
18 ssrab2 3567 . . . . . . . . 9  |-  { a  e.  A  |  ( [,] `  t ) 
C_  ( [,] `  a
) }  C_  A
195adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  A  C_  ran  F )
2018, 19syl5ss 3497 . . . . . . . 8  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  C_  ran  F )
21 simprl 755 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  -> 
t  e.  A )
22 ssid 3505 . . . . . . . . . 10  |-  ( [,] `  t )  C_  ( [,] `  t )
23 fveq2 5852 . . . . . . . . . . . 12  |-  ( a  =  t  ->  ( [,] `  a )  =  ( [,] `  t
) )
2423sseq2d 3514 . . . . . . . . . . 11  |-  ( a  =  t  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  t ) ) )
2524rspcev 3194 . . . . . . . . . 10  |-  ( ( t  e.  A  /\  ( [,] `  t ) 
C_  ( [,] `  t
) )  ->  E. a  e.  A  ( [,] `  t )  C_  ( [,] `  a ) )
2621, 22, 25sylancl 662 . . . . . . . . 9  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  E. a  e.  A  ( [,] `  t ) 
C_  ( [,] `  a
) )
27 rabn0 3787 . . . . . . . . 9  |-  ( { a  e.  A  | 
( [,] `  t
)  C_  ( [,] `  a ) }  =/=  (/)  <->  E. a  e.  A  ( [,] `  t ) 
C_  ( [,] `  a
) )
2826, 27sylibr 212 . . . . . . . 8  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  =/=  (/) )
296dyadmax 21873 . . . . . . . 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 661 . . . . . . 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 5852 . . . . . . . . . . 11  |-  ( a  =  m  ->  ( [,] `  a )  =  ( [,] `  m
) )
3231sseq2d 3514 . . . . . . . . . 10  |-  ( a  =  m  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  m ) ) )
3332elrab 3241 . . . . . . . . 9  |-  ( m  e.  { a  e.  A  |  ( [,] `  t )  C_  ( [,] `  a ) }  <-> 
( m  e.  A  /\  ( [,] `  t
)  C_  ( [,] `  m ) ) )
34 simprlr 762 . . . . . . . . . . . 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 760 . . . . . . . . . . . 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 3487 . . . . . . . . . . 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 761 . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  w  ->  ( [,] `  a )  =  ( [,] `  w
) )
3938sseq2d 3514 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  w  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  w ) ) )
4039elrab 3241 . . . . . . . . . . . . . . . . . 18  |-  ( w  e.  { a  e.  A  |  ( [,] `  t )  C_  ( [,] `  a ) }  <-> 
( w  e.  A  /\  ( [,] `  t
)  C_  ( [,] `  w ) ) )
4140imbi1i 325 . . . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( [,] `  t
)  C_  ( [,] `  w )  /\  ( [,] `  m )  C_  ( [,] `  w ) )  ->  m  =  w )  <->  ( ( [,] `  t )  C_  ( [,] `  w )  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w )
) )
45 sstr2 3493 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( [,] `  t ) 
C_  ( [,] `  m
)  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  ( [,] `  t
)  C_  ( [,] `  w ) ) )
4645ad2antll 728 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  ( [,] `  t
)  C_  ( [,] `  w ) ) )
4746ancrd 554 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  ( ( [,] `  t )  C_  ( [,] `  w )  /\  ( [,] `  m ) 
C_  ( [,] `  w
) ) ) )
4847imim1d 75 . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . 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 52 . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . 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 2848 . . . . . . . . . . . . . 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 619 . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . . 17  |-  ( z  =  m  ->  ( [,] `  z )  =  ( [,] `  m
) )
5554sseq1d 3513 . . . . . . . . . . . . . . . 16  |-  ( z  =  m  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  m )  C_  ( [,] `  w ) ) )
56 equequ1 1782 . . . . . . . . . . . . . . . 16  |-  ( z  =  m  ->  (
z  =  w  <->  m  =  w ) )
5755, 56imbi12d 320 . . . . . . . . . . . . . . 15  |-  ( z  =  m  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
5857ralbidv 2880 . . . . . . . . . . . . . 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 3243 . . . . . . . . . . . . 13  |-  ( m  e.  G  <->  ( m  e.  A  /\  A. w  e.  A  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
6137, 53, 60sylanbrc 664 . . . . . . . . . . . 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 5719 . . . . . . . . . . . . . 14  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
632, 62ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  [,]
64 ssrab2 3567 . . . . . . . . . . . . . . . . 17  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
6559, 64eqsstri 3516 . . . . . . . . . . . . . . . 16  |-  G  C_  A
6665, 14syl5ss 3497 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  C_  ( RR*  X. 
RR* ) )
672fdmi 5722 . . . . . . . . . . . . . . 15  |-  dom  [,]  =  ( RR*  X.  RR* )
6866, 67syl6sseqr 3533 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  C_  dom  [,] )
6968ad2antrr 725 . . . . . . . . . . . . 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 6129 . . . . . . . . . . . . 13  |-  ( ( Fun  [,]  /\  G  C_  dom  [,] )  ->  (
m  e.  G  -> 
( [,] `  m
)  e.  ( [,] " G ) ) )
7163, 69, 70sylancr 663 . . . . . . . . . . . 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 4235 . . . . . . . . . . 11  |-  ( ( a  e.  ( [,] `  m )  /\  ( [,] `  m )  e.  ( [,] " G
) )  ->  a  e.  U. ( [,] " G
) )
7436, 72, 73syl2anc 661 . . . . . . . . . 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 605 . . . . . . . . 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 217 . . . . . . . 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 2931 . . . . . . 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 2934 . . . . 5  |-  ( ph  ->  ( E. t  e.  A  a  e.  ( [,] `  t )  ->  a  e.  U. ( [,] " G ) ) )
8017, 79sylbid 215 . . . 4  |-  ( ph  ->  ( E. i  e.  ( [,] " A
) a  e.  i  ->  a  e.  U. ( [,] " G ) ) )
811, 80syl5bi 217 . . 3  |-  ( ph  ->  ( a  e.  U. ( [,] " A )  ->  a  e.  U. ( [,] " G ) ) )
8281ssrdv 3492 . 2  |-  ( ph  ->  U. ( [,] " A
)  C_  U. ( [,] " G ) )
83 imass2 5358 . . . 4  |-  ( G 
C_  A  ->  ( [,] " G )  C_  ( [,] " A ) )
8465, 83ax-mp 5 . . 3  |-  ( [,] " G )  C_  ( [,] " A )
85 uniss 4251 . . 3  |-  ( ( [,] " G ) 
C_  ( [,] " A
)  ->  U. ( [,] " G )  C_  U. ( [,] " A
) )
8684, 85mp1i 12 . 2  |-  ( ph  ->  U. ( [,] " G
)  C_  U. ( [,] " A ) )
8782, 86eqssd 3503 1  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   E.wrex 2792   {crab 2795    i^i cin 3457    C_ wss 3458   (/)c0 3767   ~Pcpw 3993   <.cop 4016   U.cuni 4230    X. cxp 4983   dom cdm 4985   ran crn 4986   "cima 4988   Fun wfun 5568    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   RRcr 9489   1c1 9491    + caddc 9493   RR*cxr 9625    <_ cle 9627    / cdiv 10207   2c2 10586   NN0cn0 10796   ZZcz 10865   [,]cicc 11536   ^cexp 12140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fi 7869  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-ico 11539  df-icc 11540  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-sum 13483  df-rest 14692  df-topgen 14713  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-top 19266  df-bases 19268  df-topon 19269  df-cmp 19753  df-ovol 21742
This theorem is referenced by:  dyadmbl  21875  mblfinlem1  30019  mblfinlem2  30020
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