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Theorem restmetu 18622
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 958 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  =/=  (/) )
2 psmetres2 18350 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A ) )  e.  (PsMet `  A )
)
323adant1 976 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A
) )  e.  (PsMet `  A ) )
4 oveq2 6092 . . . . . . . 8  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
54imaeq2d 5206 . . . . . . 7  |-  ( a  =  b  ->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) )  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
65cbvmptv 4303 . . . . . 6  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )
76rneqi 5099 . . . . 5  |-  ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ran  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
87metustfbas 18601 . . . 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 644 . . 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 17907 . . 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 18585 . . 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 5745 . . . 4  |-  (metUnif `  D
)  e.  _V
153elfvexd 5762 . . . . 5  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  e.  _V )
16 xpexg 4992 . . . . 5  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
1715, 15, 16syl2anc 644 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( A  X.  A )  e.  _V )
18 restval 13659 . . . 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 646 . . 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 3564 . . . . . . . . . . 11  |-  ( v  i^i  ( A  X.  A ) )  C_  ( A  X.  A
)
21 sseq1 3371 . . . . . . . . . . 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 226 . . . . . . . . . 10  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  C_  ( A  X.  A
) )
23 vex 2961 . . . . . . . . . . 11  |-  u  e. 
_V
2423elpw 3807 . . . . . . . . . 10  |-  ( u  e.  ~P ( A  X.  A )  <->  u  C_  ( A  X.  A ) )
2522, 24sylibr 205 . . . . . . . . 9  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  e.  ~P ( A  X.  A ) )
2625rexlimivw 2828 . . . . . . . 8  |-  ( E. v  e.  (metUnif `  D
) u  =  ( v  i^i  ( A  X.  A ) )  ->  u  e.  ~P ( A  X.  A
) )
2726adantl 454 . . . . . . 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 1630 . . . . . . . . 9  |-  F/ v ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )
29 nfre1 2764 . . . . . . . . 9  |-  F/ v E. v  e.  (metUnif `  D ) u  =  ( v  i^i  ( A  X.  A ) )
3028, 29nfan 1847 . . . . . . . 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 1630 . . . . . . . . . . . . 13  |-  F/ a ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )
32 nfmpt1 4301 . . . . . . . . . . . . . . 15  |-  F/_ a
( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3332nfrn 5115 . . . . . . . . . . . . . 14  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3433nfcri 2568 . . . . . . . . . . . . 13  |-  F/ a  w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3531, 34nfan 1847 . . . . . . . . . . . 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 1630 . . . . . . . . . . . 12  |-  F/ a  w  C_  v
3735, 36nfan 1847 . . . . . . . . . . 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 4301 . . . . . . . . . . . . . 14  |-  F/_ a
( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
3938nfrn 5115 . . . . . . . . . . . . 13  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
40 nfcv 2574 . . . . . . . . . . . . 13  |-  F/_ a ~P u
4139, 40nfin 3549 . . . . . . . . . . . 12  |-  F/_ a
( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )
42 nfcv 2574 . . . . . . . . . . . 12  |-  F/_ a (/)
4341, 42nfne 2697 . . . . . . . . . . 11  |-  F/ a ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  =/=  (/)
44 simplr 733 . . . . . . . . . . . . . 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 3537 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( `' D " ( 0 [,) a
) )  ->  (
w  i^i  ( A  X.  A ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
4645adantl 454 . . . . . . . . . . . . . . 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 959 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  D  e.  (PsMet `  X ) )
48 psmetf 18342 . . . . . . . . . . . . . . . . 17  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
49 ffun 5596 . . . . . . . . . . . . . . . . 17  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
50 respreima 5862 . . . . . . . . . . . . . . . . 17  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
5147, 48, 49, 504syl 20 . . . . . . . . . . . . . . . 16  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) )  =  ( ( `' D "
( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
5251ad6antr 718 . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . 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 2769 . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . 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 2961 . . . . . . . . . . . . . . 15  |-  w  e. 
_V
5756inex1 4347 . . . . . . . . . . . . . 14  |-  ( w  i^i  ( A  X.  A ) )  e. 
_V
58 eqid 2438 . . . . . . . . . . . . . . 15  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )
5958elrnmpt 5120 . . . . . . . . . . . . . 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 205 . . . . . . . . . . . 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 737 . . . . . . . . . . . . . 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 3571 . . . . . . . . . . . . . 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 3564 . . . . . . . . . . . . . 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 3804 . . . . . . . . . . . . . . . . 17  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  ~P u  =  ~P (
v  i^i  ( A  X.  A ) ) )
6867eleq2d 2505 . . . . . . . . . . . . . . . 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 3807 . . . . . . . . . . . . . . . 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 254 . . . . . . . . . . . . . . 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 3565 . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . 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 890 . . . . . . . . . . . 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 3684 . . . . . . . . . . . 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 644 . . . . . . . . . . 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 733 . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . 14  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
7978elrnmpt 5120 . . . . . . . . . . . . 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 190 . . . . . . . . . . 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 2850 . . . . . . . . . 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 3662 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  X  /\  A  =/=  (/) )  ->  X  =/=  (/) )
8483ancoms 441 . . . . . . . . . . . . . 14  |-  ( ( A  =/=  (/)  /\  A  C_  X )  ->  X  =/=  (/) )
85843adant2 977 . . . . . . . . . . . . 13  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  X  =/=  (/) )
86 metuel 18613 . . . . . . . . . . . . 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 644 . . . . . . . . . . . 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 609 . . . . . . . . . . 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 453 . . . . . . . . . 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 2852 . . . . . . . . 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 701 . . . . . . . 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 449 . . . . . . . 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 2851 . . . . . . 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 520 . . . . . 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 734 . . . . . . . . . . 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 3810 . . . . . . . . . 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 963 . . . . . . . . . . 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 4984 . . . . . . . . . . 11  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
9997, 97, 98syl2anc 644 . . . . . . . . . 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 3360 . . . . . . . . 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 3477 . . . . . . . . 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 3525 . . . . . . . 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 733 . . . . . . . . . . . 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 2439 . . . . . . . . . . . 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 5206 . . . . . . . . . . . . . 14  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
106105eqeq2d 2449 . . . . . . . . . . . . 13  |-  ( a  =  b  ->  (
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) a ) )  <-> 
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) b ) ) ) )
107106rspcev 3054 . . . . . . . . . . . 12  |-  ( ( b  e.  RR+  /\  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) b
) ) )  ->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) )
108103, 104, 107syl2anc 644 . . . . . . . . . . 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 714 . . . . . . . . . . . 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 5408 . . . . . . . . . . . 12  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
111 imaexg 5220 . . . . . . . . . . . 12  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) b ) )  e.  _V )
11278elrnmpt 5120 . . . . . . . . . . . 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 20 . . . . . . . . . . 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 225 . . . . . . . . . 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 5227 . . . . . . . . . . . . . . . 16  |-  ( `' D " ( 0 [,) b ) ) 
C_  dom  D
116 fdm 5598 . . . . . . . . . . . . . . . . 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 3398 . . . . . . . . . . . . . . 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 3688 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( X  X.  X )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  ( X  X.  X ) )  =  (/) )
121119, 120sylib 190 . . . . . . . . . . . . 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 3658 . . . . . . . . . . . . 13  |-  (/)  C_  u
123121, 122syl6eqss 3400 . . . . . . . . . . . 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 5862 . . . . . . . . . . . . . 14  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) )  =  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )
125109, 48, 49, 1244syl 20 . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . 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 3810 . . . . . . . . . . . . . 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 3385 . . . . . . . . . . . . 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 3385 . . . . . . . . . . . 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 3525 . . . . . . . . . . 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 3713 . . . . . . . . . . . 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 3714 . . . . . . . . . . . 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 3600 . . . . . . . . . . . . 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 3374 . . . . . . . . . . . 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 264 . . . . . . . . . . 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 205 . . . . . . . . . 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 3371 . . . . . . . . . . 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 3054 . . . . . . . . . 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 644 . . . . . . . . 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 3532 . . . . . . . . . . . . . 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 2961 . . . . . . . . . . . . . . . 16  |-  v  e. 
_V
1436elrnmpt 5120 . . . . . . . . . . . . . . . 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 678 . . . . . . . . . . . . . 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 439 . . . . . . . . . . . . . 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 264 . . . . . . . . . . . . 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 1593 . . . . . . . . . . . 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 3639 . . . . . . . . . . . 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 2713 . . . . . . . . . . . 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 270 . . . . . . . . . . 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 188 . . . . . . . . . 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 711 . . . . . . . . 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 2854 . . . . . . . 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 453 . . . . . . . . 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 453 . . . . . . . . 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 18613 . . . . . . . . 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 644 . . . . . . . 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 890 . . . . . . 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 3591 . . . . . . . . 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 3535 . . . . . . . . . . 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 3702 . . . . . . . . . . 11  |-  ( ( A  X.  A )  i^i  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  =  (/)
163161, 162eqtr3i 2460 . . . . . . . . . 10  |-  ( ( ( X  X.  X
)  \  ( A  X.  A ) )  i^i  ( A  X.  A
) )  =  (/)
164163uneq2i 3500 . . . . . . . . 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 3654 . . . . . . . . 9  |-  ( ( u  i^i  ( A  X.  A ) )  u.  (/) )  =  ( u  i^i  ( A  X.  A ) )
166160, 164, 1653eqtri 2462 . . . . . . . 8  |-  ( ( u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) )  i^i  ( A  X.  A ) )  =  ( u  i^i  ( A  X.  A ) )
167 df-ss 3336 . . . . . . . . 9  |-  ( u 
C_  ( A  X.  A )  <->  ( u  i^i  ( A  X.  A
) )  =  u )
16896, 167sylib 190 . . . . . . . 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 2483 . . . . . . 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 3537 . . . . . . . . 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 2449 . . . . . . . 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 3054 . . . . . . 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 644 . . . . . 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 807 . . . . 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 2438 . . . . . . 7  |-  ( v  e.  (metUnif `  D
)  |->  ( v  i^i  ( A  X.  A
) ) )  =  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) )
176175elrnmpt 5120 . . . . . 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 3804 . . . . . . . 8  |-  ( v  =  u  ->  ~P v  =  ~P u
)
179178ineq2d 3544 . . . . . . 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 2616 . . . . . 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 3094 . . . . 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 281 . . . 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 2436 . . 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 2470 . 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 2481 1  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( (metUnif `  D
)t  ( A  X.  A
) )  =  (metUnif `  ( D  |`  ( A  X.  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801    e. cmpt 4269    X. cxp 4879   `'ccnv 4880   dom cdm 4881   ran crn 4882    |` cres 4883   "cima 4884   Fun wfun 5451   -->wf 5453   ` cfv 5457  (class class class)co 6084   0cc0 8995   RR*cxr 9124   RR+crp 10617   [,)cico 10923   ↾t crest 13653  PsMetcpsmet 16690   fBascfbas 16694   filGencfg 16695  metUnifcmetu 16698
This theorem is referenced by:  reust  24349  qqhucn  24381  rrhre  24392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-rp 10618  df-ico 10927  df-rest 13655  df-psmet 16699  df-fbas 16704  df-fg 16705  df-metu 16707
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