MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dyadmbllem Unicode version

Theorem dyadmbllem 18954
Description: Lemma for dyadmbl 18955. (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 3831 . . . 4  |-  ( a  e.  U. ( [,] " A )  <->  E. i  e.  ( [,] " A
) a  e.  i )
2 iccf 10742 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
3 ffn 5389 . . . . . . 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 18946 . . . . . . . . 9  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
8 frn 5395 . . . . . . . . 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 3390 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
11 ressxr 8876 . . . . . . . . . 10  |-  RR  C_  RR*
12 xpss12 4792 . . . . . . . . . 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 3188 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
159, 14sstri 3188 . . . . . . 7  |-  ran  F  C_  ( RR*  X.  RR* )
165, 15syl6ss 3191 . . . . . 6  |-  ( ph  ->  A  C_  ( RR*  X. 
RR* ) )
17 eleq2 2344 . . . . . . 7  |-  ( i  =  ( [,] `  t
)  ->  ( a  e.  i  <->  a  e.  ( [,] `  t ) ) )
1817rexima 5757 . . . . . 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 3258 . . . . . . . . . 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 3190 . . . . . . . . 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 3197 . . . . . . . . . . 11  |-  ( [,] `  t )  C_  ( [,] `  t )
25 fveq2 5525 . . . . . . . . . . . . 13  |-  ( a  =  t  ->  ( [,] `  a )  =  ( [,] `  t
) )
2625sseq2d 3206 . . . . . . . . . . . 12  |-  ( a  =  t  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  t ) ) )
2726rspcev 2884 . . . . . . . . . . 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 3474 . . . . . . . . . 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 18953 . . . . . . . . 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 5525 . . . . . . . . . . . 12  |-  ( a  =  m  ->  ( [,] `  a )  =  ( [,] `  m
) )
3433sseq2d 3206 . . . . . . . . . . 11  |-  ( a  =  m  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  m ) ) )
3534elrab 2923 . . . . . . . . . 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 3181 . . . . . . . . . . . 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 5525 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =  w  ->  ( [,] `  a )  =  ( [,] `  w
) )
4140sseq2d 3206 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  w  ->  (
( [,] `  t
)  C_  ( [,] `  a )  <->  ( [,] `  t )  C_  ( [,] `  w ) ) )
4241elrab 2923 . . . . . . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . . . . . . . . . . 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 2623 . . . . . . . . . . . . . . 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 5525 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  m  ->  ( [,] `  z )  =  ( [,] `  m
) )
5756sseq1d 3205 . . . . . . . . . . . . . . . . 17  |-  ( z  =  m  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  m )  C_  ( [,] `  w ) ) )
58 eqeq1 2289 . . . . . . . . . . . . . . . . 17  |-  ( z  =  m  ->  (
z  =  w  <->  m  =  w ) )
5957, 58imbi12d 311 . . . . . . . . . . . . . . . 16  |-  ( z  =  m  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  m )  C_  ( [,] `  w )  ->  m  =  w ) ) )
6059ralbidv 2563 . . . . . . . . . . . . . . 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 2925 . . . . . . . . . . . . . 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 5391 . . . . . . . . . . . . . . 15  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
652, 64ax-mp 8 . . . . . . . . . . . . . 14  |-  Fun  [,]
66 ssrab2 3258 . . . . . . . . . . . . . . . . . 18  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
6761, 66eqsstri 3208 . . . . . . . . . . . . . . . . 17  |-  G  C_  A
6867, 16syl5ss 3190 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  C_  ( RR*  X. 
RR* ) )
692fdmi 5394 . . . . . . . . . . . . . . . 16  |-  dom  [,]  =  ( RR*  X.  RR* )
7068, 69syl6sseqr 3225 . . . . . . . . . . . . . . 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 5754 . . . . . . . . . . . . . 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 3832 . . . . . . . . . . . 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 2666 . . . . . . . 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 2667 . . . . 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 3185 . 2  |-  ( ph  ->  U. ( [,] " A
)  C_  U. ( [,] " G ) )
86 imass2 5049 . . . 4  |-  ( G 
C_  A  ->  ( [,] " G )  C_  ( [,] " A ) )
8767, 86ax-mp 8 . . 3  |-  ( [,] " G )  C_  ( [,] " A )
88 uniss 3848 . . 3  |-  ( ( [,] " G ) 
C_  ( [,] " A
)  ->  U. ( [,] " G )  C_  U. ( [,] " A
) )
8987, 88mp1i 11 . 2  |-  ( ph  ->  U. ( [,] " G
)  C_  U. ( [,] " A ) )
9085, 89eqssd 3196 1  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   <.cop 3643   U.cuni 3827    X. cxp 4687   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   RRcr 8736   1c1 8738    + caddc 8740   RR*cxr 8866    <_ cle 8868    / cdiv 9423   2c2 9795   NN0cn0 9965   ZZcz 10024   [,]cicc 10659   ^cexp 11104
This theorem is referenced by:  dyadmbl  18955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cmp 17114  df-ovol 18824
  Copyright terms: Public domain W3C validator