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Theorem restmetu 20278
Description: The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.)
Assertion
Ref Expression
restmetu  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( (metUnif `  D
)t  ( A  X.  A
) )  =  (metUnif `  ( D  |`  ( A  X.  A ) ) ) )

Proof of Theorem restmetu
Dummy variables  a 
b  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 988 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  =/=  (/) )
2 psmetres2 20006 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A ) )  e.  (PsMet `  A )
)
323adant1 1006 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A
) )  e.  (PsMet `  A ) )
4 oveq2 6198 . . . . . . . 8  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
54imaeq2d 5267 . . . . . . 7  |-  ( a  =  b  ->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) )  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
65cbvmptv 4481 . . . . . 6  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )
76rneqi 5164 . . . . 5  |-  ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ran  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
87metustfbas 20257 . . . 4  |-  ( ( A  =/=  (/)  /\  ( D  |`  ( A  X.  A ) )  e.  (PsMet `  A )
)  ->  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  e.  ( fBas `  ( A  X.  A ) ) )
91, 3, 8syl2anc 661 . . 3  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  e.  ( fBas `  ( A  X.  A ) ) )
10 fgval 19559 . . 3  |-  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  e.  (
fBas `  ( A  X.  A ) )  -> 
( ( A  X.  A ) filGen ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) ) )  =  { v  e.  ~P ( A  X.  A )  |  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P v )  =/=  (/) } )
119, 10syl 16 . 2  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( ( A  X.  A ) filGen ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) ) )  =  { v  e.  ~P ( A  X.  A
)  |  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P v )  =/=  (/) } )
12 metuval 20241 . . 3  |-  ( ( D  |`  ( A  X.  A ) )  e.  (PsMet `  A )  ->  (metUnif `  ( D  |`  ( A  X.  A
) ) )  =  ( ( A  X.  A ) filGen ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) ) ) )
133, 12syl 16 . 2  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  (metUnif `  ( D  |`  ( A  X.  A
) ) )  =  ( ( A  X.  A ) filGen ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) ) ) )
14 fvex 5799 . . . 4  |-  (metUnif `  D
)  e.  _V
153elfvexd 5817 . . . . 5  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  e.  _V )
16 xpexg 6607 . . . . 5  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
1715, 15, 16syl2anc 661 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( A  X.  A )  e.  _V )
18 restval 14467 . . . 4  |-  ( ( (metUnif `  D )  e.  _V  /\  ( A  X.  A )  e. 
_V )  ->  (
(metUnif `  D )t  ( A  X.  A ) )  =  ran  ( v  e.  (metUnif `  D
)  |->  ( v  i^i  ( A  X.  A
) ) ) )
1914, 17, 18sylancr 663 . . 3  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( (metUnif `  D
)t  ( A  X.  A
) )  =  ran  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) ) )
20 inss2 3669 . . . . . . . . . . 11  |-  ( v  i^i  ( A  X.  A ) )  C_  ( A  X.  A
)
21 sseq1 3475 . . . . . . . . . . 11  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  (
u  C_  ( A  X.  A )  <->  ( v  i^i  ( A  X.  A
) )  C_  ( A  X.  A ) ) )
2220, 21mpbiri 233 . . . . . . . . . 10  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  C_  ( A  X.  A
) )
23 vex 3071 . . . . . . . . . . 11  |-  u  e. 
_V
2423elpw 3964 . . . . . . . . . 10  |-  ( u  e.  ~P ( A  X.  A )  <->  u  C_  ( A  X.  A ) )
2522, 24sylibr 212 . . . . . . . . 9  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  e.  ~P ( A  X.  A ) )
2625rexlimivw 2933 . . . . . . . 8  |-  ( E. v  e.  (metUnif `  D
) u  =  ( v  i^i  ( A  X.  A ) )  ->  u  e.  ~P ( A  X.  A
) )
2726adantl 466 . . . . . . 7  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) )  ->  u  e.  ~P ( A  X.  A
) )
28 nfv 1674 . . . . . . . . 9  |-  F/ v ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )
29 nfre1 2881 . . . . . . . . 9  |-  F/ v E. v  e.  (metUnif `  D ) u  =  ( v  i^i  ( A  X.  A ) )
3028, 29nfan 1863 . . . . . . . 8  |-  F/ v ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  E. v  e.  (metUnif `  D
) u  =  ( v  i^i  ( A  X.  A ) ) )
31 nfv 1674 . . . . . . . . . . . . 13  |-  F/ a ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )
32 nfmpt1 4479 . . . . . . . . . . . . . . 15  |-  F/_ a
( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3332nfrn 5180 . . . . . . . . . . . . . 14  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3433nfcri 2606 . . . . . . . . . . . . 13  |-  F/ a  w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3531, 34nfan 1863 . . . . . . . . . . . 12  |-  F/ a ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )  /\  w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
36 nfv 1674 . . . . . . . . . . . 12  |-  F/ a  w  C_  v
3735, 36nfan 1863 . . . . . . . . . . 11  |-  F/ a ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )  /\  w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )  /\  w  C_  v )
38 nfmpt1 4479 . . . . . . . . . . . . . 14  |-  F/_ a
( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
3938nfrn 5180 . . . . . . . . . . . . 13  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
40 nfcv 2613 . . . . . . . . . . . . 13  |-  F/_ a ~P u
4139, 40nfin 3655 . . . . . . . . . . . 12  |-  F/_ a
( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )
42 nfcv 2613 . . . . . . . . . . . 12  |-  F/_ a (/)
4341, 42nfne 2779 . . . . . . . . . . 11  |-  F/ a ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  =/=  (/)
44 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
a  e.  RR+ )
45 ineq1 3643 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( `' D " ( 0 [,) a
) )  ->  (
w  i^i  ( A  X.  A ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
4645adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( w  i^i  ( A  X.  A ) )  =  ( ( `' D " ( 0 [,) a ) )  i^i  ( A  X.  A ) ) )
47 simp2 989 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  D  e.  (PsMet `  X ) )
48 psmetf 19998 . . . . . . . . . . . . . . . . 17  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
49 ffun 5659 . . . . . . . . . . . . . . . . 17  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
50 respreima 5931 . . . . . . . . . . . . . . . . 17  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
5147, 48, 49, 504syl 21 . . . . . . . . . . . . . . . 16  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) )  =  ( ( `' D "
( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
5251ad6antr 735 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) )  =  ( ( `' D " ( 0 [,) a ) )  i^i  ( A  X.  A ) ) )
5346, 52eqtr4d 2495 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( w  i^i  ( A  X.  A ) )  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )
54 rspe 2885 . . . . . . . . . . . . . 14  |-  ( ( a  e.  RR+  /\  (
w  i^i  ( A  X.  A ) )  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  ->  E. a  e.  RR+  ( w  i^i  ( A  X.  A
) )  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
5544, 53, 54syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  ->  E. a  e.  RR+  (
w  i^i  ( A  X.  A ) )  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
56 vex 3071 . . . . . . . . . . . . . . 15  |-  w  e. 
_V
5756inex1 4531 . . . . . . . . . . . . . 14  |-  ( w  i^i  ( A  X.  A ) )  e. 
_V
58 eqid 2451 . . . . . . . . . . . . . . 15  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )
5958elrnmpt 5184 . . . . . . . . . . . . . 14  |-  ( ( w  i^i  ( A  X.  A ) )  e.  _V  ->  (
( w  i^i  ( A  X.  A ) )  e.  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  <->  E. a  e.  RR+  (
w  i^i  ( A  X.  A ) )  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) ) )
6057, 59ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( w  i^i  ( A  X.  A ) )  e.  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  <->  E. a  e.  RR+  (
w  i^i  ( A  X.  A ) )  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
6155, 60sylibr 212 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( w  i^i  ( A  X.  A ) )  e.  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) ) )
62 simpllr 758 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  ->  w  C_  v )
63 ssinss1 3676 . . . . . . . . . . . . . 14  |-  ( w 
C_  v  ->  (
w  i^i  ( A  X.  A ) )  C_  v )
6462, 63syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( w  i^i  ( A  X.  A ) ) 
C_  v )
65 inss2 3669 . . . . . . . . . . . . . 14  |-  ( w  i^i  ( A  X.  A ) )  C_  ( A  X.  A
)
6665a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( w  i^i  ( A  X.  A ) ) 
C_  ( A  X.  A ) )
67 pweq 3961 . . . . . . . . . . . . . . . . 17  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  ~P u  =  ~P (
v  i^i  ( A  X.  A ) ) )
6867eleq2d 2521 . . . . . . . . . . . . . . . 16  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  (
( w  i^i  ( A  X.  A ) )  e.  ~P u  <->  ( w  i^i  ( A  X.  A
) )  e.  ~P ( v  i^i  ( A  X.  A ) ) ) )
6957elpw 3964 . . . . . . . . . . . . . . . 16  |-  ( ( w  i^i  ( A  X.  A ) )  e.  ~P ( v  i^i  ( A  X.  A ) )  <->  ( w  i^i  ( A  X.  A
) )  C_  (
v  i^i  ( A  X.  A ) ) )
7068, 69syl6bb 261 . . . . . . . . . . . . . . 15  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  (
( w  i^i  ( A  X.  A ) )  e.  ~P u  <->  ( w  i^i  ( A  X.  A
) )  C_  (
v  i^i  ( A  X.  A ) ) ) )
71 ssin 3670 . . . . . . . . . . . . . . 15  |-  ( ( ( w  i^i  ( A  X.  A ) ) 
C_  v  /\  (
w  i^i  ( A  X.  A ) )  C_  ( A  X.  A
) )  <->  ( w  i^i  ( A  X.  A
) )  C_  (
v  i^i  ( A  X.  A ) ) )
7270, 71syl6bbr 263 . . . . . . . . . . . . . 14  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  (
( w  i^i  ( A  X.  A ) )  e.  ~P u  <->  ( (
w  i^i  ( A  X.  A ) )  C_  v  /\  ( w  i^i  ( A  X.  A
) )  C_  ( A  X.  A ) ) ) )
7372ad5antlr 734 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( ( w  i^i  ( A  X.  A
) )  e.  ~P u 
<->  ( ( w  i^i  ( A  X.  A
) )  C_  v  /\  ( w  i^i  ( A  X.  A ) ) 
C_  ( A  X.  A ) ) ) )
7464, 66, 73mpbir2and 913 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( w  i^i  ( A  X.  A ) )  e.  ~P u )
75 inelcm 3831 . . . . . . . . . . . 12  |-  ( ( ( w  i^i  ( A  X.  A ) )  e.  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  /\  ( w  i^i  ( A  X.  A
) )  e.  ~P u )  ->  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) )
7661, 74, 75syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  /\  w  e. 
ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  /\  w  C_  v
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )  =/=  (/) )
77 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )  /\  w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )  /\  w  C_  v )  ->  w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )
78 eqid 2451 . . . . . . . . . . . . . 14  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
7978elrnmpt 5184 . . . . . . . . . . . . 13  |-  ( w  e.  _V  ->  (
w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  w  =  ( `' D " ( 0 [,) a ) ) ) )
8056, 79ax-mp 5 . . . . . . . . . . . 12  |-  ( w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  w  =  ( `' D " ( 0 [,) a ) ) )
8177, 80sylib 196 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )  /\  w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )  /\  w  C_  v )  ->  E. a  e.  RR+  w  =  ( `' D " ( 0 [,) a ) ) )
8237, 43, 76, 81r19.29af2 2956 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )  /\  w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )  /\  w  C_  v )  ->  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) )
83 ssn0 3768 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  X  /\  A  =/=  (/) )  ->  X  =/=  (/) )
8483ancoms 453 . . . . . . . . . . . . . 14  |-  ( ( A  =/=  (/)  /\  A  C_  X )  ->  X  =/=  (/) )
85843adant2 1007 . . . . . . . . . . . . 13  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  X  =/=  (/) )
86 metuel 20269 . . . . . . . . . . . . 13  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( v  e.  (metUnif `  D )  <->  ( v  C_  ( X  X.  X )  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  v
) ) )
8785, 47, 86syl2anc 661 . . . . . . . . . . . 12  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( v  e.  (metUnif `  D )  <->  ( v  C_  ( X  X.  X )  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  v
) ) )
8887simplbda 624 . . . . . . . . . . 11  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  ->  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  v
)
8988adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  ->  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) w 
C_  v )
9082, 89r19.29a 2958 . . . . . . . . 9  |-  ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D )
)  /\  u  =  ( v  i^i  ( A  X.  A ) ) )  ->  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) )
9190adantllr 718 . . . . . . . 8  |-  ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )  -> 
( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )  =/=  (/) )
92 simpr 461 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) )  ->  E. v  e.  (metUnif `  D ) u  =  ( v  i^i  ( A  X.  A ) ) )
9330, 91, 92r19.29af 2957 . . . . . . 7  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) )  ->  ( ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  =/=  (/) )
9427, 93jca 532 . . . . . 6  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) )  ->  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )
95 simprl 755 . . . . . . . . . . 11  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  u  e.  ~P ( A  X.  A
) )
9695elpwid 3968 . . . . . . . . . 10  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  u  C_  ( A  X.  A ) )
97 simpl3 993 . . . . . . . . . . 11  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  A  C_  X
)
98 xpss12 5043 . . . . . . . . . . 11  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
9997, 97, 98syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  ( A  X.  A )  C_  ( X  X.  X ) )
10096, 99sstrd 3464 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  u  C_  ( X  X.  X ) )
101 difssd 3582 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  ( ( X  X.  X )  \ 
( A  X.  A
) )  C_  ( X  X.  X ) )
102100, 101unssd 3630 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  C_  ( X  X.  X ) )
103 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
b  e.  RR+ )
104 eqidd 2452 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) b ) ) )
1054imaeq2d 5267 . . . . . . . . . . . . . 14  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
106105eqeq2d 2465 . . . . . . . . . . . . 13  |-  ( a  =  b  ->  (
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) a ) )  <-> 
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) b ) ) ) )
107106rspcev 3169 . . . . . . . . . . . 12  |-  ( ( b  e.  RR+  /\  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) b
) ) )  ->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) )
108103, 104, 107syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  ->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) )
10947ad4antr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  ->  D  e.  (PsMet `  X
) )
110 cnvexg 6624 . . . . . . . . . . . 12  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
111 imaexg 6615 . . . . . . . . . . . 12  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) b ) )  e.  _V )
11278elrnmpt 5184 . . . . . . . . . . . 12  |-  ( ( `' D " ( 0 [,) b ) )  e.  _V  ->  (
( `' D "
( 0 [,) b
) )  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) ) )
113109, 110, 111, 1124syl 21 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( ( `' D " ( 0 [,) b
) )  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) ) )
114108, 113mpbird 232 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( `' D "
( 0 [,) b
) )  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
115 cnvimass 5287 . . . . . . . . . . . . . . . 16  |-  ( `' D " ( 0 [,) b ) ) 
C_  dom  D
116 fdm 5661 . . . . . . . . . . . . . . . . 17  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
11748, 116syl 16 . . . . . . . . . . . . . . . 16  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
118115, 117syl5sseq 3502 . . . . . . . . . . . . . . 15  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) b ) )  C_  ( X  X.  X
) )
119109, 118syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( `' D "
( 0 [,) b
) )  C_  ( X  X.  X ) )
120 ssdif0 3835 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( X  X.  X )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  ( X  X.  X ) )  =  (/) )
121119, 120sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( ( `' D " ( 0 [,) b
) )  \  ( X  X.  X ) )  =  (/) )
122 0ss 3764 . . . . . . . . . . . . 13  |-  (/)  C_  u
123121, 122syl6eqss 3504 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( ( `' D " ( 0 [,) b
) )  \  ( X  X.  X ) ) 
C_  u )
124 respreima 5931 . . . . . . . . . . . . . 14  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) )  =  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )
125109, 48, 49, 1244syl 21 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) )  =  ( ( `' D " ( 0 [,) b ) )  i^i  ( A  X.  A ) ) )
126 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
127 simpllr 758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
v  e.  ~P u
)
128127elpwid 3968 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
v  C_  u )
129126, 128eqsstr3d 3489 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) )  C_  u )
130125, 129eqsstr3d 3489 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) 
C_  u )
131123, 130unssd 3630 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( ( ( `' D " ( 0 [,) b ) ) 
\  ( X  X.  X ) )  u.  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )  C_  u )
132 ssundif 3860 . . . . . . . . . . . 12  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  u )  C_  ( ( X  X.  X )  \  ( A  X.  A ) ) )
133 difcom 3861 . . . . . . . . . . . 12  |-  ( ( ( `' D "
( 0 [,) b
) )  \  u
)  C_  ( ( X  X.  X )  \ 
( A  X.  A
) )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  C_  u )
134 difdif2 3705 . . . . . . . . . . . . 13  |-  ( ( `' D " ( 0 [,) b ) ) 
\  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  =  ( ( ( `' D " ( 0 [,) b ) ) 
\  ( X  X.  X ) )  u.  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )
135134sseq1i 3478 . . . . . . . . . . . 12  |-  ( ( ( `' D "
( 0 [,) b
) )  \  (
( X  X.  X
)  \  ( A  X.  A ) ) ) 
C_  u  <->  ( (
( `' D "
( 0 [,) b
) )  \  ( X  X.  X ) )  u.  ( ( `' D " ( 0 [,) b ) )  i^i  ( A  X.  A ) ) ) 
C_  u )
136132, 133, 1353bitri 271 . . . . . . . . . . 11  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  <->  ( ( ( `' D " ( 0 [,) b ) ) 
\  ( X  X.  X ) )  u.  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )  C_  u )
137131, 136sylibr 212 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( `' D "
( 0 [,) b
) )  C_  (
u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) ) )
138 sseq1 3475 . . . . . . . . . . 11  |-  ( w  =  ( `' D " ( 0 [,) b
) )  ->  (
w  C_  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  <->  ( `' D " ( 0 [,) b
) )  C_  (
u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) ) ) )
139138rspcev 3169 . . . . . . . . . 10  |-  ( ( ( `' D "
( 0 [,) b
) )  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  /\  ( `' D " ( 0 [,) b ) ) 
C_  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) ) )  ->  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) w 
C_  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) ) )
140114, 137, 139syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  ->  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  (
u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) ) )
141 elin 3637 . . . . . . . . . . . . . 14  |-  ( v  e.  ( ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  <-> 
( v  e.  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  /\  v  e.  ~P u ) )
142 vex 3071 . . . . . . . . . . . . . . . 16  |-  v  e. 
_V
1436elrnmpt 5184 . . . . . . . . . . . . . . . 16  |-  ( v  e.  _V  ->  (
v  e.  ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  <->  E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) ) )
144142, 143ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( v  e.  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  <->  E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )
145144anbi1i 695 . . . . . . . . . . . . . 14  |-  ( ( v  e.  ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  /\  v  e.  ~P u )  <->  ( E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) )  /\  v  e. 
~P u ) )
146 ancom 450 . . . . . . . . . . . . . 14  |-  ( ( E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) )  /\  v  e.  ~P u
)  <->  ( v  e. 
~P u  /\  E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) ) ) )
147141, 145, 1463bitri 271 . . . . . . . . . . . . 13  |-  ( v  e.  ( ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  <-> 
( v  e.  ~P u  /\  E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) ) ) )
148147exbii 1635 . . . . . . . . . . . 12  |-  ( E. v  v  e.  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )  <->  E. v
( v  e.  ~P u  /\  E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) ) ) )
149 n0 3744 . . . . . . . . . . . 12  |-  ( ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )  =/=  (/) 
<->  E. v  v  e.  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u ) )
150 df-rex 2801 . . . . . . . . . . . 12  |-  ( E. v  e.  ~P  u E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) )  <->  E. v
( v  e.  ~P u  /\  E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) ) ) )
151148, 149, 1503bitr4i 277 . . . . . . . . . . 11  |-  ( ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )  =/=  (/) 
<->  E. v  e.  ~P  u E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
152151biimpi 194 . . . . . . . . . 10  |-  ( ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )  =/=  (/)  ->  E. v  e.  ~P  u E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
153152ad2antll 728 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  E. v  e.  ~P  u E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
154140, 153r19.29_2a 2960 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  (
u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) ) )
15585adantr 465 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  X  =/=  (/) )
15647adantr 465 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  D  e.  (PsMet `  X ) )
157 metuel 20269 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( (
u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) )  e.  (metUnif `  D
)  <->  ( ( u  u.  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  C_  ( X  X.  X
)  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) w 
C_  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) ) ) ) )
158155, 156, 157syl2anc 661 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  ( ( u  u.  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  e.  (metUnif `  D )  <->  ( ( u  u.  (
( X  X.  X
)  \  ( A  X.  A ) ) ) 
C_  ( X  X.  X )  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  (
u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) ) ) ) )
159102, 154, 158mpbir2and 913 . . . . . . 7  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  e.  (metUnif `  D
) )
160 indir 3696 . . . . . . . . 9  |-  ( ( u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) )  i^i  ( A  X.  A ) )  =  ( ( u  i^i  ( A  X.  A
) )  u.  (
( ( X  X.  X )  \  ( A  X.  A ) )  i^i  ( A  X.  A ) ) )
161 incom 3641 . . . . . . . . . . 11  |-  ( ( A  X.  A )  i^i  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  =  ( ( ( X  X.  X )  \ 
( A  X.  A
) )  i^i  ( A  X.  A ) )
162 disjdif 3849 . . . . . . . . . . 11  |-  ( ( A  X.  A )  i^i  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  =  (/)
163161, 162eqtr3i 2482 . . . . . . . . . 10  |-  ( ( ( X  X.  X
)  \  ( A  X.  A ) )  i^i  ( A  X.  A
) )  =  (/)
164163uneq2i 3605 . . . . . . . . 9  |-  ( ( u  i^i  ( A  X.  A ) )  u.  ( ( ( X  X.  X ) 
\  ( A  X.  A ) )  i^i  ( A  X.  A
) ) )  =  ( ( u  i^i  ( A  X.  A
) )  u.  (/) )
165 un0 3760 . . . . . . . . 9  |-  ( ( u  i^i  ( A  X.  A ) )  u.  (/) )  =  ( u  i^i  ( A  X.  A ) )
166160, 164, 1653eqtri 2484 . . . . . . . 8  |-  ( ( u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) )  i^i  ( A  X.  A ) )  =  ( u  i^i  ( A  X.  A ) )
167 df-ss 3440 . . . . . . . . 9  |-  ( u 
C_  ( A  X.  A )  <->  ( u  i^i  ( A  X.  A
) )  =  u )
16896, 167sylib 196 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  ( u  i^i  ( A  X.  A
) )  =  u )
169166, 168syl5req 2505 . . . . . . 7  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  u  =  ( ( u  u.  (
( X  X.  X
)  \  ( A  X.  A ) ) )  i^i  ( A  X.  A ) ) )
170 ineq1 3643 . . . . . . . . 9  |-  ( v  =  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  ->  ( v  i^i  ( A  X.  A
) )  =  ( ( u  u.  (
( X  X.  X
)  \  ( A  X.  A ) ) )  i^i  ( A  X.  A ) ) )
171170eqeq2d 2465 . . . . . . . 8  |-  ( v  =  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  ->  ( u  =  ( v  i^i  ( A  X.  A
) )  <->  u  =  ( ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  i^i  ( A  X.  A ) ) ) )
172171rspcev 3169 . . . . . . 7  |-  ( ( ( u  u.  (
( X  X.  X
)  \  ( A  X.  A ) ) )  e.  (metUnif `  D
)  /\  u  =  ( ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  i^i  ( A  X.  A ) ) )  ->  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) )
173159, 169, 172syl2anc 661 . . . . . 6  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  E. v  e.  (metUnif `  D ) u  =  ( v  i^i  ( A  X.  A ) ) )
17494, 173impbida 828 . . . . 5  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) )  <->  ( u  e.  ~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) ) )
175 eqid 2451 . . . . . . 7  |-  ( v  e.  (metUnif `  D
)  |->  ( v  i^i  ( A  X.  A
) ) )  =  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) )
176175elrnmpt 5184 . . . . . 6  |-  ( u  e.  _V  ->  (
u  e.  ran  (
v  e.  (metUnif `  D
)  |->  ( v  i^i  ( A  X.  A
) ) )  <->  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) ) )
17723, 176ax-mp 5 . . . . 5  |-  ( u  e.  ran  ( v  e.  (metUnif `  D
)  |->  ( v  i^i  ( A  X.  A
) ) )  <->  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) )
178 pweq 3961 . . . . . . . 8  |-  ( v  =  u  ->  ~P v  =  ~P u
)
179178ineq2d 3650 . . . . . . 7  |-  ( v  =  u  ->  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P v )  =  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u ) )
180179neeq1d 2725 . . . . . 6  |-  ( v  =  u  ->  (
( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P v )  =/=  (/) 
<->  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )
181180elrab 3214 . . . . 5  |-  ( u  e.  { v  e. 
~P ( A  X.  A )  |  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P v )  =/=  (/) }  <->  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )
182174, 177, 1813bitr4g 288 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( u  e. 
ran  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) )  <->  u  e.  { v  e.  ~P ( A  X.  A )  |  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P v )  =/=  (/) } ) )
183182eqrdv 2448 . . 3  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ran  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) )  =  { v  e.  ~P ( A  X.  A )  |  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P v )  =/=  (/) } )
18419, 183eqtrd 2492 . 2  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( (metUnif `  D
)t  ( A  X.  A
) )  =  {
v  e.  ~P ( A  X.  A )  |  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P v )  =/=  (/) } )
18511, 13, 1843eqtr4rd 2503 1  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( (metUnif `  D
)t  ( A  X.  A
) )  =  (metUnif `  ( D  |`  ( A  X.  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2644   E.wrex 2796   {crab 2799   _Vcvv 3068    \ cdif 3423    u. cun 3424    i^i cin 3425    C_ wss 3426   (/)c0 3735   ~Pcpw 3958    |-> cmpt 4448    X. cxp 4936   `'ccnv 4937   dom cdm 4938   ran crn 4939    |` cres 4940   "cima 4941   Fun wfun 5510   -->wf 5512   ` cfv 5516  (class class class)co 6190   0cc0 9383   RR*cxr 9518   RR+crp 11092   [,)cico 11403   ↾t crest 14461  PsMetcpsmet 17909   fBascfbas 17913   filGencfg 17914  metUnifcmetu 17917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-rp 11093  df-ico 11407  df-rest 14463  df-psmet 17918  df-fbas 17923  df-fg 17924  df-metu 17926
This theorem is referenced by:  reust  21001  qqhucn  26555
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