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Theorem restmetu 18570
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 957 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  =/=  (/) )
2 psmetres2 18298 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A ) )  e.  (PsMet `  A )
)
323adant1 975 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A
) )  e.  (PsMet `  A ) )
4 oveq2 6048 . . . . . . . 8  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
54imaeq2d 5162 . . . . . . 7  |-  ( a  =  b  ->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) )  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
65cbvmptv 4260 . . . . . 6  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )
76rneqi 5055 . . . . 5  |-  ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ran  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
87metustfbas 18549 . . . 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 643 . . 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 17855 . . 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 18533 . . 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 5701 . . . 4  |-  (metUnif `  D
)  e.  _V
153elfvexd 5718 . . . . 5  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  e.  _V )
16 xpexg 4948 . . . . 5  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
1715, 15, 16syl2anc 643 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( A  X.  A )  e.  _V )
18 restval 13609 . . . 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 645 . . 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 3522 . . . . . . . . . . 11  |-  ( v  i^i  ( A  X.  A ) )  C_  ( A  X.  A
)
21 sseq1 3329 . . . . . . . . . . 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 225 . . . . . . . . . 10  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  C_  ( A  X.  A
) )
23 vex 2919 . . . . . . . . . . 11  |-  u  e. 
_V
2423elpw 3765 . . . . . . . . . 10  |-  ( u  e.  ~P ( A  X.  A )  <->  u  C_  ( A  X.  A ) )
2522, 24sylibr 204 . . . . . . . . 9  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  e.  ~P ( A  X.  A ) )
2625rexlimivw 2786 . . . . . . . 8  |-  ( E. v  e.  (metUnif `  D
) u  =  ( v  i^i  ( A  X.  A ) )  ->  u  e.  ~P ( A  X.  A
) )
2726adantl 453 . . . . . . 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 1626 . . . . . . . . 9  |-  F/ v ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )
29 nfre1 2722 . . . . . . . . 9  |-  F/ v E. v  e.  (metUnif `  D ) u  =  ( v  i^i  ( A  X.  A ) )
3028, 29nfan 1842 . . . . . . . 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 1626 . . . . . . . . . . . . 13  |-  F/ a ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )
32 nfmpt1 4258 . . . . . . . . . . . . . . 15  |-  F/_ a
( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3332nfrn 5071 . . . . . . . . . . . . . 14  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3433nfcri 2534 . . . . . . . . . . . . 13  |-  F/ a  w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3531, 34nfan 1842 . . . . . . . . . . . 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 1626 . . . . . . . . . . . 12  |-  F/ a  w  C_  v
3735, 36nfan 1842 . . . . . . . . . . 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 4258 . . . . . . . . . . . . . 14  |-  F/_ a
( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
3938nfrn 5071 . . . . . . . . . . . . 13  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
40 nfcv 2540 . . . . . . . . . . . . 13  |-  F/_ a ~P u
4139, 40nfin 3507 . . . . . . . . . . . 12  |-  F/_ a
( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )
42 nfcv 2540 . . . . . . . . . . . 12  |-  F/_ a (/)
4341, 42nfne 2658 . . . . . . . . . . 11  |-  F/ a ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  =/=  (/)
44 simplr 732 . . . . . . . . . . . . . 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 3495 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( `' D " ( 0 [,) a
) )  ->  (
w  i^i  ( A  X.  A ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
4645adantl 453 . . . . . . . . . . . . . . 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 958 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  D  e.  (PsMet `  X ) )
48 psmetf 18290 . . . . . . . . . . . . . . . . . 18  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
49 ffun 5552 . . . . . . . . . . . . . . . . . 18  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
50 respreima 5818 . . . . . . . . . . . . . . . . . 18  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
5148, 49, 503syl 19 . . . . . . . . . . . . . . . . 17  |-  ( D  e.  (PsMet `  X
)  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
5247, 51syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) )  =  ( ( `' D "
( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
5352ad6antr 717 . . . . . . . . . . . . . . 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 ) ) )
5446, 53eqtr4d 2439 . . . . . . . . . . . . . 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
) ) )
55 rspe 2727 . . . . . . . . . . . . . 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 ) ) )
5644, 54, 55syl2anc 643 . . . . . . . . . . . . 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 ) ) )
57 vex 2919 . . . . . . . . . . . . . . 15  |-  w  e. 
_V
5857inex1 4304 . . . . . . . . . . . . . 14  |-  ( w  i^i  ( A  X.  A ) )  e. 
_V
59 eqid 2404 . . . . . . . . . . . . . . 15  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )
6059elrnmpt 5076 . . . . . . . . . . . . . 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 ) ) ) )
6158, 60ax-mp 8 . . . . . . . . . . . . 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 ) ) )
6256, 61sylibr 204 . . . . . . . . . . . 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 ) ) ) )
63 simpllr 736 . . . . . . . . . . . . . 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 )
64 ssinss1 3529 . . . . . . . . . . . . . 14  |-  ( w 
C_  v  ->  (
w  i^i  ( A  X.  A ) )  C_  v )
6563, 64syl 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 )
66 inss2 3522 . . . . . . . . . . . . . 14  |-  ( w  i^i  ( A  X.  A ) )  C_  ( A  X.  A
)
6766a1i 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 ) )
68 pweq 3762 . . . . . . . . . . . . . . . . 17  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  ~P u  =  ~P (
v  i^i  ( A  X.  A ) ) )
6968eleq2d 2471 . . . . . . . . . . . . . . . 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 ) ) ) )
7058elpw 3765 . . . . . . . . . . . . . . . 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 ) ) )
7169, 70syl6bb 253 . . . . . . . . . . . . . . 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 ) ) ) )
72 ssin 3523 . . . . . . . . . . . . . . 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 ) ) )
7371, 72syl6bbr 255 . . . . . . . . . . . . . 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 ) ) ) )
7473ad5antlr 716 . . . . . . . . . . . . 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 ) ) ) )
7565, 67, 74mpbir2and 889 . . . . . . . . . . . 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 )
76 inelcm 3642 . . . . . . . . . . . 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 )  =/=  (/) )
7762, 75, 76syl2anc 643 . . . . . . . . . . 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 )  =/=  (/) )
78 simplr 732 . . . . . . . . . . . 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
) ) ) )
79 eqid 2404 . . . . . . . . . . . . . 14  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
8079elrnmpt 5076 . . . . . . . . . . . . 13  |-  ( w  e.  _V  ->  (
w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  w  =  ( `' D " ( 0 [,) a ) ) ) )
8157, 80ax-mp 8 . . . . . . . . . . . 12  |-  ( w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  w  =  ( `' D " ( 0 [,) a ) ) )
8278, 81sylib 189 . . . . . . . . . . 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 ) ) )
8337, 43, 77, 82r19.29af2 2808 . . . . . . . . . 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 )  =/=  (/) )
84 ssn0 3620 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  X  /\  A  =/=  (/) )  ->  X  =/=  (/) )
8584ancoms 440 . . . . . . . . . . . . . 14  |-  ( ( A  =/=  (/)  /\  A  C_  X )  ->  X  =/=  (/) )
86853adant2 976 . . . . . . . . . . . . 13  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  X  =/=  (/) )
87 metuel 18561 . . . . . . . . . . . . 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
) ) )
8886, 47, 87syl2anc 643 . . . . . . . . . . . 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
) ) )
8988simplbda 608 . . . . . . . . . . 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
)
9089adantr 452 . . . . . . . . . 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 )
9183, 90r19.29a 2810 . . . . . . . . 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 )  =/=  (/) )
9291adantllr 700 . . . . . . . 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 )  =/=  (/) )
93 simpr 448 . . . . . . . 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 ) ) )
9430, 92, 93r19.29af 2809 . . . . . . 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 )  =/=  (/) )
9527, 94jca 519 . . . . . 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 )  =/=  (/) ) )
96 simprl 733 . . . . . . . . . . 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
) )
9796elpwid 3768 . . . . . . . . . 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 ) )
98 simpl3 962 . . . . . . . . . . 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
)
99 xpss12 4940 . . . . . . . . . . 11  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
10098, 98, 99syl2anc 643 . . . . . . . . . 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 ) )
10197, 100sstrd 3318 . . . . . . . . 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 ) )
102 difssd 3435 . . . . . . . . 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 ) )
103101, 102unssd 3483 . . . . . . . 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 ) )
104 simplr 732 . . . . . . . . . . . 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+ )
105 eqidd 2405 . . . . . . . . . . . 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 ) ) )
1064imaeq2d 5162 . . . . . . . . . . . . . 14  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
107106eqeq2d 2415 . . . . . . . . . . . . 13  |-  ( a  =  b  ->  (
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) a ) )  <-> 
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) b ) ) ) )
108107rspcev 3012 . . . . . . . . . . . 12  |-  ( ( b  e.  RR+  /\  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) b
) ) )  ->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) )
109104, 105, 108syl2anc 643 . . . . . . . . . . 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
) ) )
11047ad4antr 713 . . . . . . . . . . . 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
) )
111 cnvexg 5364 . . . . . . . . . . . . 13  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
112 imaexg 5176 . . . . . . . . . . . . 13  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) b ) )  e.  _V )
11379elrnmpt 5076 . . . . . . . . . . . . 13  |-  ( ( `' 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
) ) ) )
114111, 112, 1133syl 19 . . . . . . . . . . . 12  |-  ( D  e.  (PsMet `  X
)  ->  ( ( `' D " ( 0 [,) b ) )  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) ) )
115110, 114syl 16 . . . . . . . . . . 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
) ) ) )
116109, 115mpbird 224 . . . . . . . . . 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 ) ) ) )
117 cnvimass 5183 . . . . . . . . . . . . . . . 16  |-  ( `' D " ( 0 [,) b ) ) 
C_  dom  D
118 fdm 5554 . . . . . . . . . . . . . . . . 17  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
11948, 118syl 16 . . . . . . . . . . . . . . . 16  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
120117, 119syl5sseq 3356 . . . . . . . . . . . . . . 15  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) b ) )  C_  ( X  X.  X
) )
121110, 120syl 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 ) )
122 ssdif0 3646 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( X  X.  X )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  ( X  X.  X ) )  =  (/) )
123121, 122sylib 189 . . . . . . . . . . . . 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 ) )  =  (/) )
124 0ss 3616 . . . . . . . . . . . . 13  |-  (/)  C_  u
125123, 124syl6eqss 3358 . . . . . . . . . . . 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 )
126 respreima 5818 . . . . . . . . . . . . . . 15  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) )  =  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )
12748, 49, 1263syl 19 . . . . . . . . . . . . . 14  |-  ( D  e.  (PsMet `  X
)  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) )  =  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )
128110, 127syl 16 . . . . . . . . . . . . 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 ) ) )
129 simpr 448 . . . . . . . . . . . . . 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 ) ) )
130 simpllr 736 . . . . . . . . . . . . . . 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
)
131130elpwid 3768 . . . . . . . . . . . . . 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 )
132129, 131eqsstr3d 3343 . . . . . . . . . . . . 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 )
133128, 132eqsstr3d 3343 . . . . . . . . . . . 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 )
134125, 133unssd 3483 . . . . . . . . . . 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 )
135 ssundif 3671 . . . . . . . . . . . 12  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  u )  C_  ( ( X  X.  X )  \  ( A  X.  A ) ) )
136 difcom 3672 . . . . . . . . . . . 12  |-  ( ( ( `' D "
( 0 [,) b
) )  \  u
)  C_  ( ( X  X.  X )  \ 
( A  X.  A
) )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  C_  u )
137 difdif2 3558 . . . . . . . . . . . . 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 ) ) )
138137sseq1i 3332 . . . . . . . . . . . 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 )
139135, 136, 1383bitri 263 . . . . . . . . . . 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 )
140134, 139sylibr 204 . . . . . . . . . 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 ) ) ) )
141 sseq1 3329 . . . . . . . . . . 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 ) ) ) ) )
142141rspcev 3012 . . . . . . . . . 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 ) ) ) )
143116, 140, 142syl2anc 643 . . . . . . . . 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 ) ) ) )
144 elin 3490 . . . . . . . . . . . . . 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 ) )
145 vex 2919 . . . . . . . . . . . . . . . 16  |-  v  e. 
_V
1466elrnmpt 5076 . . . . . . . . . . . . . . . 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
) ) ) )
147145, 146ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( v  e.  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  <->  E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )
148147anbi1i 677 . . . . . . . . . . . . . 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 ) )
149 ancom 438 . . . . . . . . . . . . . 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 ) ) ) )
150144, 148, 1493bitri 263 . . . . . . . . . . . . 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 ) ) ) )
151150exbii 1589 . . . . . . . . . . . 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 ) ) ) )
152 n0 3597 . . . . . . . . . . . 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 ) )
153 df-rex 2672 . . . . . . . . . . . 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 ) ) ) )
154151, 152, 1533bitr4i 269 . . . . . . . . . . 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 ) ) )
155154biimpi 187 . . . . . . . . . 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 ) ) )
156155ad2antll 710 . . . . . . . . 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 ) ) )
157143, 156r19.29_2a 2812 . . . . . . . 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 ) ) ) )
15886adantr 452 . . . . . . . . 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  =/=  (/) )
15947adantr 452 . . . . . . . . 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 ) )
160 metuel 18561 . . . . . . . . 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 ) ) ) ) ) )
161158, 159, 160syl2anc 643 . . . . . . . 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 ) ) ) ) ) )
162103, 157, 161mpbir2and 889 . . . . . . 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
) )
163 indir 3549 . . . . . . . . 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 ) ) )
164 incom 3493 . . . . . . . . . . 11  |-  ( ( A  X.  A )  i^i  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  =  ( ( ( X  X.  X )  \ 
( A  X.  A
) )  i^i  ( A  X.  A ) )
165 disjdif 3660 . . . . . . . . . . 11  |-  ( ( A  X.  A )  i^i  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  =  (/)
166164, 165eqtr3i 2426 . . . . . . . . . 10  |-  ( ( ( X  X.  X
)  \  ( A  X.  A ) )  i^i  ( A  X.  A
) )  =  (/)
167166uneq2i 3458 . . . . . . . . 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.  (/) )
168 un0 3612 . . . . . . . . 9  |-  ( ( u  i^i  ( A  X.  A ) )  u.  (/) )  =  ( u  i^i  ( A  X.  A ) )
169163, 167, 1683eqtri 2428 . . . . . . . 8  |-  ( ( u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) )  i^i  ( A  X.  A ) )  =  ( u  i^i  ( A  X.  A ) )
170 df-ss 3294 . . . . . . . . 9  |-  ( u 
C_  ( A  X.  A )  <->  ( u  i^i  ( A  X.  A
) )  =  u )
17197, 170sylib 189 . . . . . . . 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 )
172169, 171syl5req 2449 . . . . . . 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 ) ) )
173 ineq1 3495 . . . . . . . . 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 ) ) )
174173eqeq2d 2415 . . . . . . . 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 ) ) ) )
175174rspcev 3012 . . . . . . 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 ) ) )
176162, 172, 175syl2anc 643 . . . . . 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 ) ) )
17795, 176impbida 806 . . . . 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 )  =/=  (/) ) ) )
178 eqid 2404 . . . . . . 7  |-  ( v  e.  (metUnif `  D
)  |->  ( v  i^i  ( A  X.  A
) ) )  =  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) )
179178elrnmpt 5076 . . . . . 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 ) ) ) )
18023, 179ax-mp 8 . . . . 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 ) ) )
181 pweq 3762 . . . . . . . 8  |-  ( v  =  u  ->  ~P v  =  ~P u
)
182181ineq2d 3502 . . . . . . 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 ) )
183182neeq1d 2580 . . . . . 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 )  =/=  (/) ) )
184183elrab 3052 . . . . 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 )  =/=  (/) ) )
185177, 180, 1843bitr4g 280 . . . 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 )  =/=  (/) } ) )
186185eqrdv 2402 . . 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 )  =/=  (/) } )
18719, 186eqtrd 2436 . 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 )  =/=  (/) } )
18811, 13, 1873eqtr4rd 2447 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 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   {crab 2670   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946   RR*cxr 9075   RR+crp 10568   [,)cico 10874   ↾t crest 13603  PsMetcpsmet 16640   fBascfbas 16644   filGencfg 16645  metUnifcmetu 16648
This theorem is referenced by:  reust  24297  qqhucn  24329  rrhre  24340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-rp 10569  df-ico 10878  df-rest 13605  df-psmet 16649  df-fbas 16654  df-fg 16655  df-metu 16657
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