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Theorem dyadmbllem 19007
Description: Lemma for dyadmbl 19008. (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 3868 . . . 4  |-  ( a  e.  U. ( [,] " A )  <->  E. i  e.  ( [,] " A
) a  e.  i )
2 iccf 10789 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5427 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  [,]  Fn  ( RR*  X.  RR* )
)
42, 3ax-mp 8 . . . . . 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 18999 . . . . . . . . 9  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
8 frn 5433 . . . . . . . . 9  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
97, 8ax-mp 8 . . . . . . . 8  |-  ran  F  C_  (  <_  i^i  ( RR  X.  RR ) )
10 inss2 3424 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
11 ressxr 8921 . . . . . . . . . 10  |-  RR  C_  RR*
12 xpss12 4829 . . . . . . . . . 10  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
1311, 11, 12mp2an 653 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
1410, 13sstri 3222 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
159, 14sstri 3222 . . . . . . 7  |-  ran  F  C_  ( RR*  X.  RR* )
165, 15syl6ss 3225 . . . . . 6  |-  ( ph  ->  A  C_  ( RR*  X. 
RR* ) )
17 eleq2 2377 . . . . . . 7  |-  ( i  =  ( [,] `  t
)  ->  ( a  e.  i  <->  a  e.  ( [,] `  t ) ) )
1817rexima 5798 . . . . . 6  |-  ( ( [,]  Fn  ( RR*  X. 
RR* )  /\  A  C_  ( RR*  X.  RR* )
)  ->  ( E. i  e.  ( [,] " A ) a  e.  i  <->  E. t  e.  A  a  e.  ( [,] `  t ) ) )
194, 16, 18sylancr 644 . . . . 5  |-  ( ph  ->  ( E. i  e.  ( [,] " A
) a  e.  i  <->  E. t  e.  A  a  e.  ( [,] `  t ) ) )
20 ssrab2 3292 . . . . . . . . . 10  |-  { a  e.  A  |  ( [,] `  t ) 
C_  ( [,] `  a
) }  C_  A
215adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  A  C_  ran  F )
2220, 21syl5ss 3224 . . . . . . . . 9  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  C_  ran  F )
23 simprl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  -> 
t  e.  A )
24 ssid 3231 . . . . . . . . . . 11  |-  ( [,] `  t )  C_  ( [,] `  t )
25 fveq2 5563 . . . . . . . . . . . . 13  |-  ( a  =  t  ->  ( [,] `  a )  =  ( [,] `  t
) )
2625sseq2d 3240 . . . . . . . . . . . 12  |-  ( a  =  t  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  t ) ) )
2726rspcev 2918 . . . . . . . . . . 11  |-  ( ( t  e.  A  /\  ( [,] `  t ) 
C_  ( [,] `  t
) )  ->  E. a  e.  A  ( [,] `  t )  C_  ( [,] `  a ) )
2823, 24, 27sylancl 643 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  E. a  e.  A  ( [,] `  t ) 
C_  ( [,] `  a
) )
29 rabn0 3508 . . . . . . . . . 10  |-  ( { a  e.  A  | 
( [,] `  t
)  C_  ( [,] `  a ) }  =/=  (/)  <->  E. a  e.  A  ( [,] `  t ) 
C_  ( [,] `  a
) )
3028, 29sylibr 203 . . . . . . . . 9  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  ->  { a  e.  A  |  ( [,] `  t
)  C_  ( [,] `  a ) }  =/=  (/) )
316dyadmax 19006 . . . . . . . . 9  |-  ( ( { 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 ) )
3222, 30, 31syl2anc 642 . . . . . . . 8  |-  ( (
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 ) )
33 fveq2 5563 . . . . . . . . . . . 12  |-  ( a  =  m  ->  ( [,] `  a )  =  ( [,] `  m
) )
3433sseq2d 3240 . . . . . . . . . . 11  |-  ( a  =  m  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  m ) ) )
3534elrab 2957 . . . . . . . . . 10  |-  ( m  e.  { a  e.  A  |  ( [,] `  t )  C_  ( [,] `  a ) }  <-> 
( m  e.  A  /\  ( [,] `  t
)  C_  ( [,] `  m ) ) )
36 simprlr 739 . . . . . . . . . . . . 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 ) ) )  ->  ( [,] `  t
)  C_  ( [,] `  m ) )
37 simplrr 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 ) ) )  ->  a  e.  ( [,] `  t ) )
3836, 37sseldd 3215 . . . . . . . . . . . 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.  ( [,] `  m ) )
39 simprll 738 . . . . . . . . . . . . . 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 ) ) )  ->  m  e.  A
)
40 fveq2 5563 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =  w  ->  ( [,] `  a )  =  ( [,] `  w
) )
4140sseq2d 3240 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  w  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  w ) ) )
4241elrab 2957 . . . . . . . . . . . . . . . . . . 19  |-  ( w  e.  { a  e.  A  |  ( [,] `  t )  C_  ( [,] `  a ) }  <-> 
( w  e.  A  /\  ( [,] `  t
)  C_  ( [,] `  w ) ) )
4342imbi1i 315 . . . . . . . . . . . . . . . . . 18  |-  ( ( 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 433 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  e.  A  /\  ( [,] `  t
)  C_  ( [,] `  w ) )  -> 
( ( [,] `  m
)  C_  ( [,] `  w )  ->  m  =  w ) )  <->  ( w  e.  A  ->  ( ( [,] `  t ) 
C_  ( [,] `  w
)  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) ) )
4543, 44bitri 240 . . . . . . . . . . . . . . . . 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 ) ) ) )
46 impexp 433 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( [,] `  t
)  C_  ( [,] `  w )  /\  ( [,] `  m )  C_  ( [,] `  w ) )  ->  m  =  w )  <->  ( ( [,] `  t )  C_  ( [,] `  w )  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w )
) )
47 sstr2 3220 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( [,] `  t ) 
C_  ( [,] `  m
)  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  ( [,] `  t
)  C_  ( [,] `  w ) ) )
4847ad2antll 709 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  ( [,] `  t
)  C_  ( [,] `  w ) ) )
4948ancrd 537 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
t  e.  A  /\  a  e.  ( [,] `  t ) ) )  /\  ( m  e.  A  /\  ( [,] `  t )  C_  ( [,] `  m ) ) )  ->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  ( ( [,] `  t )  C_  ( [,] `  w )  /\  ( [,] `  m ) 
C_  ( [,] `  w
) ) ) )
5049imim1d 69 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 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 ) ) )
5146, 50syl5bir 209 . . . . . . . . . . . . . . . . . 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 ) ) )
5251imim2d 48 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 ) ) ) )
5345, 52syl5bi 208 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) ) ) )
5453ralimdv2 2657 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )
5554impr 602 . . . . . . . . . . . . . 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 ) )
56 fveq2 5563 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  m  ->  ( [,] `  z )  =  ( [,] `  m
) )
5756sseq1d 3239 . . . . . . . . . . . . . . . . 17  |-  ( z  =  m  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  m )  C_  ( [,] `  w ) ) )
58 eqeq1 2322 . . . . . . . . . . . . . . . . 17  |-  ( z  =  m  ->  (
z  =  w  <->  m  =  w ) )
5957, 58imbi12d 311 . . . . . . . . . . . . . . . 16  |-  ( z  =  m  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
6059ralbidv 2597 . . . . . . . . . . . . . . 15  |-  ( z  =  m  ->  ( A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  A. w  e.  A  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
61 dyadmbl.2 . . . . . . . . . . . . . . 15  |-  G  =  { z  e.  A  |  A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }
6260, 61elrab2 2959 . . . . . . . . . . . . . 14  |-  ( m  e.  G  <->  ( m  e.  A  /\  A. w  e.  A  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
6339, 55, 62sylanbrc 645 . . . . . . . . . . . . 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.  G
)
64 ffun 5429 . . . . . . . . . . . . . . 15  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
652, 64ax-mp 8 . . . . . . . . . . . . . 14  |-  Fun  [,]
66 ssrab2 3292 . . . . . . . . . . . . . . . . . 18  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
6761, 66eqsstri 3242 . . . . . . . . . . . . . . . . 17  |-  G  C_  A
6867, 16syl5ss 3224 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  C_  ( RR*  X. 
RR* ) )
692fdmi 5432 . . . . . . . . . . . . . . . 16  |-  dom  [,]  =  ( RR*  X.  RR* )
7068, 69syl6sseqr 3259 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  C_  dom  [,] )
7170ad2antrr 706 . . . . . . . . . . . . . 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 ) ) )  ->  G  C_  dom  [,] )
72 funfvima2 5795 . . . . . . . . . . . . . 14  |-  ( ( Fun  [,]  /\  G  C_  dom  [,] )  ->  (
m  e.  G  -> 
( [,] `  m
)  e.  ( [,] " G ) ) )
7365, 71, 72sylancr 644 . . . . . . . . . . . . 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.  G  ->  ( [,] `  m )  e.  ( [,] " G ) ) )
7463, 73mpd 14 . . . . . . . . . . . 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 ) )
75 elunii 3869 . . . . . . . . . . . 12  |-  ( ( a  e.  ( [,] `  m )  /\  ( [,] `  m )  e.  ( [,] " G
) )  ->  a  e.  U. ( [,] " G
) )
7638, 74, 75syl2anc 642 . . . . . . . . . . 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.  U. ( [,] " G ) )
7776exp32 588 . . . . . . . . . 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
) ) ) )
7835, 77syl5bi 208 . . . . . . . . 9  |-  ( (
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
) ) ) )
7978rexlimdv 2700 . . . . . . . 8  |-  ( (
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 ) ) )
8032, 79mpd 14 . . . . . . 7  |-  ( (
ph  /\  ( t  e.  A  /\  a  e.  ( [,] `  t
) ) )  -> 
a  e.  U. ( [,] " G ) )
8180expr 598 . . . . . 6  |-  ( (
ph  /\  t  e.  A )  ->  (
a  e.  ( [,] `  t )  ->  a  e.  U. ( [,] " G
) ) )
8281rexlimdva 2701 . . . . 5  |-  ( ph  ->  ( E. t  e.  A  a  e.  ( [,] `  t )  ->  a  e.  U. ( [,] " G ) ) )
8319, 82sylbid 206 . . . 4  |-  ( ph  ->  ( E. i  e.  ( [,] " A
) a  e.  i  ->  a  e.  U. ( [,] " G ) ) )
841, 83syl5bi 208 . . 3  |-  ( ph  ->  ( a  e.  U. ( [,] " A )  ->  a  e.  U. ( [,] " G ) ) )
8584ssrdv 3219 . 2  |-  ( ph  ->  U. ( [,] " A
)  C_  U. ( [,] " G ) )
86 imass2 5086 . . . 4  |-  ( G 
C_  A  ->  ( [,] " G )  C_  ( [,] " A ) )
8767, 86ax-mp 8 . . 3  |-  ( [,] " G )  C_  ( [,] " A )
88 uniss 3885 . . 3  |-  ( ( [,] " G ) 
C_  ( [,] " A
)  ->  U. ( [,] " G )  C_  U. ( [,] " A
) )
8987, 88mp1i 11 . 2  |-  ( ph  ->  U. ( [,] " G
)  C_  U. ( [,] " A ) )
9085, 89eqssd 3230 1  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   E.wrex 2578   {crab 2581    i^i cin 3185    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   <.cop 3677   U.cuni 3864    X. cxp 4724   dom cdm 4726   ran crn 4727   "cima 4729   Fun wfun 5286    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   RRcr 8781   1c1 8783    + caddc 8785   RR*cxr 8911    <_ cle 8913    / cdiv 9468   2c2 9840   NN0cn0 10012   ZZcz 10071   [,]cicc 10706   ^cexp 11151
This theorem is referenced by:  dyadmbl  19008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-ico 10709  df-icc 10710  df-fz 10830  df-fzo 10918  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-clim 12009  df-sum 12206  df-rest 13376  df-topgen 13393  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-top 16692  df-bases 16694  df-topon 16695  df-cmp 17170  df-ovol 18877
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