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Theorem restmetu 21663
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 1030 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  =/=  (/) )
2 psmetres2 21408 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A ) )  e.  (PsMet `  A )
)
323adant1 1048 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A
) )  e.  (PsMet `  A ) )
4 oveq2 6316 . . . . . . . 8  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
54imaeq2d 5174 . . . . . . 7  |-  ( a  =  b  ->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) )  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
65cbvmptv 4488 . . . . . 6  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )
76rneqi 5067 . . . . 5  |-  ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ran  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
87metustfbas 21650 . . . 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 673 . . 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 20963 . . 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 17 . 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 21642 . . 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 17 . 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 5889 . . . 4  |-  (metUnif `  D
)  e.  _V
153elfvexd 5907 . . . . 5  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  e.  _V )
16 xpexg 6612 . . . . 5  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
1715, 15, 16syl2anc 673 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( A  X.  A )  e.  _V )
18 restval 15403 . . . 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 676 . . 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 3644 . . . . . . . . . . 11  |-  ( v  i^i  ( A  X.  A ) )  C_  ( A  X.  A
)
21 sseq1 3439 . . . . . . . . . . 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 241 . . . . . . . . . 10  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  C_  ( A  X.  A
) )
23 vex 3034 . . . . . . . . . . 11  |-  u  e. 
_V
2423elpw 3948 . . . . . . . . . 10  |-  ( u  e.  ~P ( A  X.  A )  <->  u  C_  ( A  X.  A ) )
2522, 24sylibr 217 . . . . . . . . 9  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  e.  ~P ( A  X.  A ) )
2625rexlimivw 2869 . . . . . . . 8  |-  ( E. v  e.  (metUnif `  D
) u  =  ( v  i^i  ( A  X.  A ) )  ->  u  e.  ~P ( A  X.  A
) )
2726adantl 473 . . . . . . 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 1769 . . . . . . . . . . . 12  |-  F/ a ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )
29 nfmpt1 4485 . . . . . . . . . . . . . 14  |-  F/_ a
( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3029nfrn 5083 . . . . . . . . . . . . 13  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3130nfcri 2606 . . . . . . . . . . . 12  |-  F/ a  w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3228, 31nfan 2031 . . . . . . . . . . 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 ) ) ) )
33 nfv 1769 . . . . . . . . . . 11  |-  F/ a  w  C_  v
3432, 33nfan 2031 . . . . . . . . . 10  |-  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 )
35 nfmpt1 4485 . . . . . . . . . . . . 13  |-  F/_ a
( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
3635nfrn 5083 . . . . . . . . . . . 12  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
37 nfcv 2612 . . . . . . . . . . . 12  |-  F/_ a ~P u
3836, 37nfin 3630 . . . . . . . . . . 11  |-  F/_ a
( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )
39 nfcv 2612 . . . . . . . . . . 11  |-  F/_ a (/)
4038, 39nfne 2742 . . . . . . . . . 10  |-  F/ a ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  =/=  (/)
41 simplr 770 . . . . . . . . . . . . 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
) ) )  -> 
a  e.  RR+ )
42 ineq1 3618 . . . . . . . . . . . . . . 15  |-  ( w  =  ( `' D " ( 0 [,) a
) )  ->  (
w  i^i  ( A  X.  A ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
4342adantl 473 . . . . . . . . . . . . . 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 " ( 0 [,) a ) )  i^i  ( A  X.  A ) ) )
44 simp2 1031 . . . . . . . . . . . . . . . 16  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  D  e.  (PsMet `  X ) )
45 psmetf 21400 . . . . . . . . . . . . . . . 16  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
46 ffun 5742 . . . . . . . . . . . . . . . 16  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
47 respreima 6024 . . . . . . . . . . . . . . . 16  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
4844, 45, 46, 474syl 19 . . . . . . . . . . . . . . 15  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) )  =  ( ( `' D "
( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
4948ad6antr 750 . . . . . . . . . . . . . 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
) ) )  -> 
( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) )  =  ( ( `' D " ( 0 [,) a ) )  i^i  ( A  X.  A ) ) )
5043, 49eqtr4d 2508 . . . . . . . . . . . . 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 ) )  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )
51 rspe 2844 . . . . . . . . . . . . 13  |-  ( ( 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 ) ) )
5241, 50, 51syl2anc 673 . . . . . . . . . . . 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
) ) )  ->  E. a  e.  RR+  (
w  i^i  ( A  X.  A ) )  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
53 vex 3034 . . . . . . . . . . . . . 14  |-  w  e. 
_V
5453inex1 4537 . . . . . . . . . . . . 13  |-  ( w  i^i  ( A  X.  A ) )  e. 
_V
55 eqid 2471 . . . . . . . . . . . . . 14  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )
5655elrnmpt 5087 . . . . . . . . . . . . 13  |-  ( ( 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 ) ) ) )
5754, 56ax-mp 5 . . . . . . . . . . . 12  |-  ( ( 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 ) ) )
5852, 57sylibr 217 . . . . . . . . . . 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
) ) )  -> 
( w  i^i  ( A  X.  A ) )  e.  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) ) )
59 simpllr 777 . . . . . . . . . . . . 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  C_  v )
60 ssinss1 3651 . . . . . . . . . . . . 13  |-  ( w 
C_  v  ->  (
w  i^i  ( A  X.  A ) )  C_  v )
6159, 60syl 17 . . . . . . . . . . . 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 ) ) 
C_  v )
62 inss2 3644 . . . . . . . . . . . . 13  |-  ( w  i^i  ( A  X.  A ) )  C_  ( A  X.  A
)
6362a1i 11 . . . . . . . . . . . 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 ) ) 
C_  ( A  X.  A ) )
64 pweq 3945 . . . . . . . . . . . . . . . 16  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  ~P u  =  ~P (
v  i^i  ( A  X.  A ) ) )
6564eleq2d 2534 . . . . . . . . . . . . . . 15  |-  ( 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 ) ) ) )
6654elpw 3948 . . . . . . . . . . . . . . 15  |-  ( ( 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 ) ) )
6765, 66syl6bb 269 . . . . . . . . . . . . . 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  i^i  ( A  X.  A ) ) ) )
68 ssin 3645 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) )
6967, 68syl6bbr 271 . . . . . . . . . . . . 13  |-  ( 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 ) ) ) )
7069ad5antlr 749 . . . . . . . . . . . 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 
<->  ( ( w  i^i  ( A  X.  A
) )  C_  v  /\  ( w  i^i  ( A  X.  A ) ) 
C_  ( A  X.  A ) ) ) )
7161, 63, 70mpbir2and 936 . . . . . . . . . . 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
) ) )  -> 
( w  i^i  ( A  X.  A ) )  e.  ~P u )
72 inelcm 3823 . . . . . . . . . . 11  |-  ( ( ( 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 )  =/=  (/) )
7358, 71, 72syl2anc 673 . . . . . . . . . 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
)  /\  a  e.  RR+ )  /\  w  =  ( `' D "
( 0 [,) a
) ) )  -> 
( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )  =/=  (/) )
74 simplr 770 . . . . . . . . . . 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 )  ->  w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )
75 eqid 2471 . . . . . . . . . . . . 13  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
7675elrnmpt 5087 . . . . . . . . . . . 12  |-  ( w  e.  _V  ->  (
w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  w  =  ( `' D " ( 0 [,) a ) ) ) )
7753, 76ax-mp 5 . . . . . . . . . . 11  |-  ( w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  w  =  ( `' D " ( 0 [,) a ) ) )
7874, 77sylib 201 . . . . . . . . . 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 )  ->  E. a  e.  RR+  w  =  ( `' D " ( 0 [,) a ) ) )
7934, 40, 73, 78r19.29af2 2915 . . . . . . . . 9  |-  ( ( ( ( ( ( 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 )  =/=  (/) )
80 ssn0 3770 . . . . . . . . . . . . . 14  |-  ( ( A  C_  X  /\  A  =/=  (/) )  ->  X  =/=  (/) )
8180ancoms 460 . . . . . . . . . . . . 13  |-  ( ( A  =/=  (/)  /\  A  C_  X )  ->  X  =/=  (/) )
82813adant2 1049 . . . . . . . . . . . 12  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  X  =/=  (/) )
83 metuel 21657 . . . . . . . . . . . 12  |-  ( ( 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
) ) )
8482, 44, 83syl2anc 673 . . . . . . . . . . 11  |-  ( ( 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
) ) )
8584simplbda 636 . . . . . . . . . 10  |-  ( ( ( 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
)
8685adantr 472 . . . . . . . . 9  |-  ( ( ( ( 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 )
8779, 86r19.29a 2918 . . . . . . . 8  |-  ( ( ( ( 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 )  =/=  (/) )
8887r19.29an 2917 . . . . . . 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 )  =/=  (/) )
8927, 88jca 541 . . . . . 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 )  =/=  (/) ) )
90 simprl 772 . . . . . . . . . . 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
) )
9190elpwid 3952 . . . . . . . . . 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 ) )
92 simpl3 1035 . . . . . . . . . . 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
)
93 xpss12 4945 . . . . . . . . . . 11  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
9492, 92, 93syl2anc 673 . . . . . . . . . 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 ) )
9591, 94sstrd 3428 . . . . . . . . 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 ) )
96 difssd 3550 . . . . . . . . 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 ) )
9795, 96unssd 3601 . . . . . . . 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 ) )
98 simplr 770 . . . . . . . . . . . 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+ )
99 eqidd 2472 . . . . . . . . . . . 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 ) ) )
1004imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
101100eqeq2d 2481 . . . . . . . . . . . . 13  |-  ( a  =  b  ->  (
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) a ) )  <-> 
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) b ) ) ) )
102101rspcev 3136 . . . . . . . . . . . 12  |-  ( ( b  e.  RR+  /\  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) b
) ) )  ->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) )
10398, 99, 102syl2anc 673 . . . . . . . . . . 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
) ) )
10444ad4antr 746 . . . . . . . . . . . 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
) )
105 cnvexg 6758 . . . . . . . . . . . 12  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
106 imaexg 6749 . . . . . . . . . . . 12  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) b ) )  e.  _V )
10775elrnmpt 5087 . . . . . . . . . . . 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
) ) ) )
108104, 105, 106, 1074syl 19 . . . . . . . . . . 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
) ) ) )
109103, 108mpbird 240 . . . . . . . . . 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 ) ) ) )
110 cnvimass 5194 . . . . . . . . . . . . . . . 16  |-  ( `' D " ( 0 [,) b ) ) 
C_  dom  D
111 fdm 5745 . . . . . . . . . . . . . . . . 17  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
11245, 111syl 17 . . . . . . . . . . . . . . . 16  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
113110, 112syl5sseq 3466 . . . . . . . . . . . . . . 15  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) b ) )  C_  ( X  X.  X
) )
114104, 113syl 17 . . . . . . . . . . . . . 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 ) )
115 ssdif0 3741 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( X  X.  X )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  ( X  X.  X ) )  =  (/) )
116114, 115sylib 201 . . . . . . . . . . . . 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 ) )  =  (/) )
117 0ss 3766 . . . . . . . . . . . . 13  |-  (/)  C_  u
118116, 117syl6eqss 3468 . . . . . . . . . . . 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 )
119 respreima 6024 . . . . . . . . . . . . . 14  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) )  =  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )
120104, 45, 46, 1194syl 19 . . . . . . . . . . . . 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 ) ) )
121 simpr 468 . . . . . . . . . . . . . 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 ) ) )
122 simpllr 777 . . . . . . . . . . . . . . 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
)
123122elpwid 3952 . . . . . . . . . . . . . 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 )
124121, 123eqsstr3d 3453 . . . . . . . . . . . . 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 )
125120, 124eqsstr3d 3453 . . . . . . . . . . . 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 )
126118, 125unssd 3601 . . . . . . . . . . 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 )
127 ssundif 3842 . . . . . . . . . . . 12  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  u )  C_  ( ( X  X.  X )  \  ( A  X.  A ) ) )
128 difcom 3843 . . . . . . . . . . . 12  |-  ( ( ( `' D "
( 0 [,) b
) )  \  u
)  C_  ( ( X  X.  X )  \ 
( A  X.  A
) )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  C_  u )
129 difdif2 3691 . . . . . . . . . . . . 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 ) ) )
130129sseq1i 3442 . . . . . . . . . . . 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 )
131127, 128, 1303bitri 279 . . . . . . . . . . 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 )
132126, 131sylibr 217 . . . . . . . . . 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 ) ) ) )
133 sseq1 3439 . . . . . . . . . . 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 ) ) ) ) )
134133rspcev 3136 . . . . . . . . . 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 ) ) ) )
135109, 132, 134syl2anc 673 . . . . . . . . 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 ) ) ) )
136 elin 3608 . . . . . . . . . . . . . 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 ) )
137 vex 3034 . . . . . . . . . . . . . . . 16  |-  v  e. 
_V
1386elrnmpt 5087 . . . . . . . . . . . . . . . 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
) ) ) )
139137, 138ax-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
) ) )
140139anbi1i 709 . . . . . . . . . . . . . 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 ) )
141 ancom 457 . . . . . . . . . . . . . 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 ) ) ) )
142136, 140, 1413bitri 279 . . . . . . . . . . . . 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 ) ) ) )
143142exbii 1726 . . . . . . . . . . . 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 ) ) ) )
144 n0 3732 . . . . . . . . . . . 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 ) )
145 df-rex 2762 . . . . . . . . . . . 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 ) ) ) )
146143, 144, 1453bitr4i 285 . . . . . . . . . . 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 ) ) )
147146biimpi 199 . . . . . . . . . 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 ) ) )
148147ad2antll 743 . . . . . . . . 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 ) ) )
149135, 148r19.29vva 2920 . . . . . . . 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 ) ) ) )
15082adantr 472 . . . . . . . . 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  =/=  (/) )
15144adantr 472 . . . . . . . . 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 ) )
152 metuel 21657 . . . . . . . . 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 ) ) ) ) ) )
153150, 151, 152syl2anc 673 . . . . . . . 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 ) ) ) ) ) )
15497, 149, 153mpbir2and 936 . . . . . . 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
) )
155 indir 3682 . . . . . . . . 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 ) ) )
156 incom 3616 . . . . . . . . . . 11  |-  ( ( A  X.  A )  i^i  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  =  ( ( ( X  X.  X )  \ 
( A  X.  A
) )  i^i  ( A  X.  A ) )
157 disjdif 3830 . . . . . . . . . . 11  |-  ( ( A  X.  A )  i^i  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  =  (/)
158156, 157eqtr3i 2495 . . . . . . . . . 10  |-  ( ( ( X  X.  X
)  \  ( A  X.  A ) )  i^i  ( A  X.  A
) )  =  (/)
159158uneq2i 3576 . . . . . . . . 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.  (/) )
160 un0 3762 . . . . . . . . 9  |-  ( ( u  i^i  ( A  X.  A ) )  u.  (/) )  =  ( u  i^i  ( A  X.  A ) )
161155, 159, 1603eqtri 2497 . . . . . . . 8  |-  ( ( u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) )  i^i  ( A  X.  A ) )  =  ( u  i^i  ( A  X.  A ) )
162 df-ss 3404 . . . . . . . . 9  |-  ( u 
C_  ( A  X.  A )  <->  ( u  i^i  ( A  X.  A
) )  =  u )
16391, 162sylib 201 . . . . . . . 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 )
164161, 163syl5req 2518 . . . . . . 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 ) ) )
165 ineq1 3618 . . . . . . . . 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 ) ) )
166165eqeq2d 2481 . . . . . . . 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 ) ) ) )
167166rspcev 3136 . . . . . . 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 ) ) )
168154, 164, 167syl2anc 673 . . . . . 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 ) ) )
16989, 168impbida 850 . . . . 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 )  =/=  (/) ) ) )
170 eqid 2471 . . . . . . 7  |-  ( v  e.  (metUnif `  D
)  |->  ( v  i^i  ( A  X.  A
) ) )  =  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) )
171170elrnmpt 5087 . . . . . 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 ) ) ) )
17223, 171ax-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 ) ) )
173 pweq 3945 . . . . . . . 8  |-  ( v  =  u  ->  ~P v  =  ~P u
)
174173ineq2d 3625 . . . . . . 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 ) )
175174neeq1d 2702 . . . . . 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 )  =/=  (/) ) )
176175elrab 3184 . . . . 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 )  =/=  (/) ) )
177169, 172, 1763bitr4g 296 . . . 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 )  =/=  (/) } ) )
178177eqrdv 2469 . . 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 )  =/=  (/) } )
17919, 178eqtrd 2505 . 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 )  =/=  (/) } )
18011, 13, 1793eqtr4rd 2516 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   {crab 2760   _Vcvv 3031    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   RR*cxr 9692   RR+crp 11325   [,)cico 11662   ↾t crest 15397  PsMetcpsmet 19031   fBascfbas 19035   filGencfg 19036  metUnifcmetu 19038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-rp 11326  df-ico 11666  df-rest 15399  df-psmet 19039  df-fbas 19044  df-fg 19045  df-metu 19046
This theorem is referenced by:  reust  22418  qqhucn  28870
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