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Theorem dyadmbllem 21054
Description: Lemma for dyadmbl 21055. (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 4090 . . . 4  |-  ( a  e.  U. ( [,] " A )  <->  E. i  e.  ( [,] " A
) a  e.  i )
2 iccf 11380 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5554 . . . . . . 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 21046 . . . . . . . . 9  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
8 frn 5560 . . . . . . . . 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 3566 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
11 rexpssxrxp 9420 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
1210, 11sstri 3360 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
139, 12sstri 3360 . . . . . . 7  |-  ran  F  C_  ( RR*  X.  RR* )
145, 13syl6ss 3363 . . . . . 6  |-  ( ph  ->  A  C_  ( RR*  X. 
RR* ) )
15 eleq2 2499 . . . . . . 7  |-  ( i  =  ( [,] `  t
)  ->  ( a  e.  i  <->  a  e.  ( [,] `  t ) ) )
1615rexima 5951 . . . . . 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 3432 . . . . . . . . 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 3362 . . . . . . . 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 3370 . . . . . . . . . 10  |-  ( [,] `  t )  C_  ( [,] `  t )
23 fveq2 5686 . . . . . . . . . . . 12  |-  ( a  =  t  ->  ( [,] `  a )  =  ( [,] `  t
) )
2423sseq2d 3379 . . . . . . . . . . 11  |-  ( a  =  t  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  t ) ) )
2524rspcev 3068 . . . . . . . . . 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 3652 . . . . . . . . 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 21053 . . . . . . . 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 5686 . . . . . . . . . . 11  |-  ( a  =  m  ->  ( [,] `  a )  =  ( [,] `  m
) )
3231sseq2d 3379 . . . . . . . . . 10  |-  ( a  =  m  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  m ) ) )
3332elrab 3112 . . . . . . . . 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 3352 . . . . . . . . . . 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 5686 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  w  ->  ( [,] `  a )  =  ( [,] `  w
) )
3938sseq2d 3379 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  w  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  w ) ) )
4039elrab 3112 . . . . . . . . . . . . . . . . . 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 3358 . . . . . . . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . 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 5686 . . . . . . . . . . . . . . . . 17  |-  ( z  =  m  ->  ( [,] `  z )  =  ( [,] `  m
) )
5554sseq1d 3378 . . . . . . . . . . . . . . . 16  |-  ( z  =  m  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  m )  C_  ( [,] `  w ) ) )
56 equequ1 1736 . . . . . . . . . . . . . . . 16  |-  ( z  =  m  ->  (
z  =  w  <->  m  =  w ) )
5755, 56imbi12d 320 . . . . . . . . . . . . . . 15  |-  ( z  =  m  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
5857ralbidv 2730 . . . . . . . . . . . . . 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 3114 . . . . . . . . . . . . 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 5556 . . . . . . . . . . . . . 14  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
632, 62ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  [,]
64 ssrab2 3432 . . . . . . . . . . . . . . . . 17  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
6559, 64eqsstri 3381 . . . . . . . . . . . . . . . 16  |-  G  C_  A
6665, 14syl5ss 3362 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  C_  ( RR*  X. 
RR* ) )
672fdmi 5559 . . . . . . . . . . . . . . 15  |-  dom  [,]  =  ( RR*  X.  RR* )
6866, 67syl6sseqr 3398 . . . . . . . . . . . . . 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 5948 . . . . . . . . . . . . 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 4091 . . . . . . . . . . 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 2835 . . . . . . 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 2837 . . . . 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 3357 . 2  |-  ( ph  ->  U. ( [,] " A
)  C_  U. ( [,] " G ) )
83 imass2 5199 . . . 4  |-  ( G 
C_  A  ->  ( [,] " G )  C_  ( [,] " A ) )
8465, 83ax-mp 5 . . 3  |-  ( [,] " G )  C_  ( [,] " A )
85 uniss 4107 . . 3  |-  ( ( [,] " G ) 
C_  ( [,] " A
)  ->  U. ( [,] " G )  C_  U. ( [,] " A
) )
8684, 85mp1i 12 . 2  |-  ( ph  ->  U. ( [,] " G
)  C_  U. ( [,] " A ) )
8782, 86eqssd 3368 1  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E.wrex 2711   {crab 2714    i^i cin 3322    C_ wss 3323   (/)c0 3632   ~Pcpw 3855   <.cop 3878   U.cuni 4086    X. cxp 4833   dom cdm 4835   ran crn 4836   "cima 4838   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   RRcr 9273   1c1 9275    + caddc 9277   RR*cxr 9409    <_ cle 9411    / cdiv 9985   2c2 10363   NN0cn0 10571   ZZcz 10638   [,]cicc 11295   ^cexp 11857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-rest 14353  df-topgen 14374  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-top 18478  df-bases 18480  df-topon 18481  df-cmp 18965  df-ovol 20923
This theorem is referenced by:  dyadmbl  21055  mblfinlem1  28381  mblfinlem2  28382
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