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Theorem restmetu 21216
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 996 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  =/=  (/) )
2 psmetres2 20944 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A ) )  e.  (PsMet `  A )
)
323adant1 1014 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( D  |`  ( A  X.  A
) )  e.  (PsMet `  A ) )
4 oveq2 6304 . . . . . . . 8  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
54imaeq2d 5347 . . . . . . 7  |-  ( a  =  b  ->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) )  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
65cbvmptv 4548 . . . . . 6  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )
76rneqi 5239 . . . . 5  |-  ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ran  ( b  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
87metustfbas 21195 . . . 4  |-  ( ( A  =/=  (/)  /\  ( D  |`  ( A  X.  A ) )  e.  (PsMet `  A )
)  ->  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  e.  ( fBas `  ( A  X.  A ) ) )
91, 3, 8syl2anc 661 . . 3  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  e.  ( fBas `  ( A  X.  A ) ) )
10 fgval 20497 . . 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 21179 . . 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 5882 . . . 4  |-  (metUnif `  D
)  e.  _V
153elfvexd 5900 . . . . 5  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  A  e.  _V )
16 xpexg 6601 . . . . 5  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
1715, 15, 16syl2anc 661 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( A  X.  A )  e.  _V )
18 restval 14844 . . . 4  |-  ( ( (metUnif `  D )  e.  _V  /\  ( A  X.  A )  e. 
_V )  ->  (
(metUnif `  D )t  ( A  X.  A ) )  =  ran  ( v  e.  (metUnif `  D
)  |->  ( v  i^i  ( A  X.  A
) ) ) )
1914, 17, 18sylancr 663 . . 3  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( (metUnif `  D
)t  ( A  X.  A
) )  =  ran  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) ) )
20 inss2 3715 . . . . . . . . . . 11  |-  ( v  i^i  ( A  X.  A ) )  C_  ( A  X.  A
)
21 sseq1 3520 . . . . . . . . . . 11  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  (
u  C_  ( A  X.  A )  <->  ( v  i^i  ( A  X.  A
) )  C_  ( A  X.  A ) ) )
2220, 21mpbiri 233 . . . . . . . . . 10  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  C_  ( A  X.  A
) )
23 vex 3112 . . . . . . . . . . 11  |-  u  e. 
_V
2423elpw 4021 . . . . . . . . . 10  |-  ( u  e.  ~P ( A  X.  A )  <->  u  C_  ( A  X.  A ) )
2522, 24sylibr 212 . . . . . . . . 9  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  u  e.  ~P ( A  X.  A ) )
2625rexlimivw 2946 . . . . . . . 8  |-  ( E. v  e.  (metUnif `  D
) u  =  ( v  i^i  ( A  X.  A ) )  ->  u  e.  ~P ( A  X.  A
) )
2726adantl 466 . . . . . . 7  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) )  ->  u  e.  ~P ( A  X.  A
) )
28 nfv 1708 . . . . . . . . . . . 12  |-  F/ a ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  v  e.  (metUnif `  D ) )  /\  u  =  ( v  i^i  ( A  X.  A
) ) )
29 nfmpt1 4546 . . . . . . . . . . . . . 14  |-  F/_ a
( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3029nfrn 5255 . . . . . . . . . . . . 13  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3130nfcri 2612 . . . . . . . . . . . 12  |-  F/ a  w  e.  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
3228, 31nfan 1929 . . . . . . . . . . 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 1708 . . . . . . . . . . 11  |-  F/ a  w  C_  v
3432, 33nfan 1929 . . . . . . . . . 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 4546 . . . . . . . . . . . . 13  |-  F/_ a
( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
3635nfrn 5255 . . . . . . . . . . . 12  |-  F/_ a ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )
37 nfcv 2619 . . . . . . . . . . . 12  |-  F/_ a ~P u
3836, 37nfin 3701 . . . . . . . . . . 11  |-  F/_ a
( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )
39 nfcv 2619 . . . . . . . . . . 11  |-  F/_ a (/)
4038, 39nfne 2788 . . . . . . . . . 10  |-  F/ a ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  =/=  (/)
41 simplr 755 . . . . . . . . . . . . 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 3689 . . . . . . . . . . . . . . 15  |-  ( w  =  ( `' D " ( 0 [,) a
) )  ->  (
w  i^i  ( A  X.  A ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
4342adantl 466 . . . . . . . . . . . . . 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 997 . . . . . . . . . . . . . . . 16  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  D  e.  (PsMet `  X ) )
45 psmetf 20936 . . . . . . . . . . . . . . . 16  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
46 ffun 5739 . . . . . . . . . . . . . . . 16  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
47 respreima 6017 . . . . . . . . . . . . . . . 16  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) )  =  ( ( `' D " ( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
4844, 45, 46, 474syl 21 . . . . . . . . . . . . . . 15  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) )  =  ( ( `' D "
( 0 [,) a
) )  i^i  ( A  X.  A ) ) )
4948ad6antr 735 . . . . . . . . . . . . . 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 2501 . . . . . . . . . . . . 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 2915 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 3112 . . . . . . . . . . . . . 14  |-  w  e. 
_V
5453inex1 4597 . . . . . . . . . . . . 13  |-  ( w  i^i  ( A  X.  A ) )  e. 
_V
55 eqid 2457 . . . . . . . . . . . . . 14  |-  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )
5655elrnmpt 5259 . . . . . . . . . . . . 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 212 . . . . . . . . . . 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 760 . . . . . . . . . . . . 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 3722 . . . . . . . . . . . . 13  |-  ( w 
C_  v  ->  (
w  i^i  ( A  X.  A ) )  C_  v )
6159, 60syl 16 . . . . . . . . . . . 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 3715 . . . . . . . . . . . . 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 4018 . . . . . . . . . . . . . . . 16  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  ~P u  =  ~P (
v  i^i  ( A  X.  A ) ) )
6564eleq2d 2527 . . . . . . . . . . . . . . 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 4021 . . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . . 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 3716 . . . . . . . . . . . . . 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 263 . . . . . . . . . . . . 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 734 . . . . . . . . . . . 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 922 . . . . . . . . . . 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 3884 . . . . . . . . . . 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 661 . . . . . . . . . 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 755 . . . . . . . . . . 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 2457 . . . . . . . . . . . . 13  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
7675elrnmpt 5259 . . . . . . . . . . . 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 196 . . . . . . . . . 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 2995 . . . . . . . . 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 3827 . . . . . . . . . . . . . 14  |-  ( ( A  C_  X  /\  A  =/=  (/) )  ->  X  =/=  (/) )
8180ancoms 453 . . . . . . . . . . . . 13  |-  ( ( A  =/=  (/)  /\  A  C_  X )  ->  X  =/=  (/) )
82813adant2 1015 . . . . . . . . . . . 12  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  X  =/=  (/) )
83 metuel 21207 . . . . . . . . . . . 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 661 . . . . . . . . . . 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 624 . . . . . . . . . 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 465 . . . . . . . . 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 2999 . . . . . . . 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 2998 . . . . . . 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 532 . . . . . 6  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  E. v  e.  (metUnif `  D )
u  =  ( v  i^i  ( A  X.  A ) ) )  ->  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )
90 simprl 756 . . . . . . . . . . 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 4025 . . . . . . . . . 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 1001 . . . . . . . . . . 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 5117 . . . . . . . . . . 11  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
9492, 92, 93syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  ( A  X.  A )  C_  ( X  X.  X ) )
9591, 94sstrd 3509 . . . . . . . . 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 3628 . . . . . . . . 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 3676 . . . . . . . 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 755 . . . . . . . . . . . 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 2458 . . . . . . . . . . . 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 5347 . . . . . . . . . . . . . 14  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
101100eqeq2d 2471 . . . . . . . . . . . . 13  |-  ( a  =  b  ->  (
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) a ) )  <-> 
( `' D "
( 0 [,) b
) )  =  ( `' D " ( 0 [,) b ) ) ) )
102101rspcev 3210 . . . . . . . . . . . 12  |-  ( ( b  e.  RR+  /\  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) b
) ) )  ->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) )
10398, 99, 102syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  ->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) )
10444ad4antr 731 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  ->  D  e.  (PsMet `  X
) )
105 cnvexg 6745 . . . . . . . . . . . 12  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
106 imaexg 6736 . . . . . . . . . . . 12  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) b ) )  e.  _V )
10775elrnmpt 5259 . . . . . . . . . . . 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 21 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( ( `' D " ( 0 [,) b
) )  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) a
) ) ) )
109103, 108mpbird 232 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( `' D "
( 0 [,) b
) )  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
110 cnvimass 5367 . . . . . . . . . . . . . . . 16  |-  ( `' D " ( 0 [,) b ) ) 
C_  dom  D
111 fdm 5741 . . . . . . . . . . . . . . . . 17  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
11245, 111syl 16 . . . . . . . . . . . . . . . 16  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
113110, 112syl5sseq 3547 . . . . . . . . . . . . . . 15  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) b ) )  C_  ( X  X.  X
) )
114104, 113syl 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 ) )
115 ssdif0 3888 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( X  X.  X )  <->  ( ( `' D " ( 0 [,) b ) ) 
\  ( X  X.  X ) )  =  (/) )
116114, 115sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( ( `' D " ( 0 [,) b
) )  \  ( X  X.  X ) )  =  (/) )
117 0ss 3823 . . . . . . . . . . . . 13  |-  (/)  C_  u
118116, 117syl6eqss 3549 . . . . . . . . . . . 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 6017 . . . . . . . . . . . . . 14  |-  ( Fun 
D  ->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) )  =  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )
120104, 45, 46, 1194syl 21 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) )  =  ( ( `' D " ( 0 [,) b ) )  i^i  ( A  X.  A ) ) )
121 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
122 simpllr 760 . . . . . . . . . . . . . . 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 4025 . . . . . . . . . . . . . 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 3534 . . . . . . . . . . . . 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 3534 . . . . . . . . . . . 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 3676 . . . . . . . . . . 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 3914 . . . . . . . . . . . 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 3915 . . . . . . . . . . . 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 3762 . . . . . . . . . . . . 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 3523 . . . . . . . . . . . 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 271 . . . . . . . . . . 11  |-  ( ( `' D " ( 0 [,) b ) ) 
C_  ( u  u.  ( ( X  X.  X )  \  ( A  X.  A ) ) )  <->  ( ( ( `' D " ( 0 [,) b ) ) 
\  ( X  X.  X ) )  u.  ( ( `' D " ( 0 [,) b
) )  i^i  ( A  X.  A ) ) )  C_  u )
132126, 131sylibr 212 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  -> 
( `' D "
( 0 [,) b
) )  C_  (
u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) ) )
133 sseq1 3520 . . . . . . . . . . 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 3210 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ( ( ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  /\  v  e.  ~P u )  /\  b  e.  RR+ )  /\  v  =  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) b
) ) )  ->  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  (
u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) ) )
136 elin 3683 . . . . . . . . . . . . . 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 3112 . . . . . . . . . . . . . . . 16  |-  v  e. 
_V
1386elrnmpt 5259 . . . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . . 14  |-  ( ( v  e.  ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  /\  v  e.  ~P u )  <->  ( E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) )  /\  v  e. 
~P u ) )
141 ancom 450 . . . . . . . . . . . . . 14  |-  ( ( E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) )  /\  v  e.  ~P u
)  <->  ( v  e. 
~P u  /\  E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) ) ) )
142136, 140, 1413bitri 271 . . . . . . . . . . . . 13  |-  ( v  e.  ( ran  (
a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P u )  <-> 
( v  e.  ~P u  /\  E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) b ) ) ) )
143142exbii 1668 . . . . . . . . . . . 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 3803 . . . . . . . . . . . 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 2813 . . . . . . . . . . . 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 277 . . . . . . . . . . 11  |-  ( ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )  =/=  (/) 
<->  E. v  e.  ~P  u E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
147146biimpi 194 . . . . . . . . . 10  |-  ( ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) ) "
( 0 [,) a
) ) )  i^i 
~P u )  =/=  (/)  ->  E. v  e.  ~P  u E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
148147ad2antll 728 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  E. v  e.  ~P  u E. b  e.  RR+  v  =  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) b ) ) )
149135, 148r19.29_2a 3001 . . . . . . . 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 465 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  X  =/=  (/) )
15144adantr 465 . . . . . . . . 9  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  D  e.  (PsMet `  X ) )
152 metuel 21207 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  ( ( u  u.  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  e.  (metUnif `  D )  <->  ( ( u  u.  (
( X  X.  X
)  \  ( A  X.  A ) ) ) 
C_  ( X  X.  X )  /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  (
u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) ) ) ) )
15497, 149, 153mpbir2and 922 . . . . . . 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 3753 . . . . . . . . 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 3687 . . . . . . . . . . 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 3903 . . . . . . . . . . 11  |-  ( ( A  X.  A )  i^i  ( ( X  X.  X )  \ 
( A  X.  A
) ) )  =  (/)
158156, 157eqtr3i 2488 . . . . . . . . . 10  |-  ( ( ( X  X.  X
)  \  ( A  X.  A ) )  i^i  ( A  X.  A
) )  =  (/)
159158uneq2i 3651 . . . . . . . . 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 3819 . . . . . . . . 9  |-  ( ( u  i^i  ( A  X.  A ) )  u.  (/) )  =  ( u  i^i  ( A  X.  A ) )
161155, 159, 1603eqtri 2490 . . . . . . . 8  |-  ( ( u  u.  ( ( X  X.  X ) 
\  ( A  X.  A ) ) )  i^i  ( A  X.  A ) )  =  ( u  i^i  ( A  X.  A ) )
162 df-ss 3485 . . . . . . . . 9  |-  ( u 
C_  ( A  X.  A )  <->  ( u  i^i  ( A  X.  A
) )  =  u )
16391, 162sylib 196 . . . . . . . 8  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  ( u  i^i  ( A  X.  A
) )  =  u )
164161, 163syl5req 2511 . . . . . . 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 3689 . . . . . . . . 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 2471 . . . . . . . 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 3210 . . . . . . 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 661 . . . . . 6  |-  ( ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  /\  ( u  e. 
~P ( A  X.  A )  /\  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A
) ) " (
0 [,) a ) ) )  i^i  ~P u )  =/=  (/) ) )  ->  E. v  e.  (metUnif `  D ) u  =  ( v  i^i  ( A  X.  A ) ) )
16989, 168impbida 832 . . . . 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 2457 . . . . . . 7  |-  ( v  e.  (metUnif `  D
)  |->  ( v  i^i  ( A  X.  A
) ) )  =  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) )
171170elrnmpt 5259 . . . . . 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 4018 . . . . . . . 8  |-  ( v  =  u  ->  ~P v  =  ~P u
)
174173ineq2d 3696 . . . . . . 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 2734 . . . . . 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 3257 . . . . 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 288 . . . 4  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( u  e. 
ran  ( v  e.  (metUnif `  D )  |->  ( v  i^i  ( A  X.  A ) ) )  <->  u  e.  { v  e.  ~P ( A  X.  A )  |  ( ran  ( a  e.  RR+  |->  ( `' ( D  |`  ( A  X.  A ) )
" ( 0 [,) a ) ) )  i^i  ~P v )  =/=  (/) } ) )
178177eqrdv 2454 . . 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 2498 . 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 2509 1  |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( (metUnif `  D
)t  ( A  X.  A
) )  =  (metUnif `  ( D  |`  ( A  X.  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   E.wrex 2808   {crab 2811   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   RR*cxr 9644   RR+crp 11245   [,)cico 11556   ↾t crest 14838  PsMetcpsmet 18529   fBascfbas 18533   filGencfg 18534  metUnifcmetu 18537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-rp 11246  df-ico 11560  df-rest 14840  df-psmet 18538  df-fbas 18543  df-fg 18544  df-metu 18546
This theorem is referenced by:  reust  21939  qqhucn  28134
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