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Theorem metustss 20128
Description: Range of the elements of the filter base generated by the metric  D. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metustss  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X
) )
Distinct variable groups:    D, a    X, a    A, a    F, a

Proof of Theorem metustss
StepHypRef Expression
1 metust.1 . . . 4  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
2 cnvimass 5188 . . . . . . . . 9  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
3 psmetf 19881 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
4 fdm 5562 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
53, 4syl 16 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
62, 5syl5sseq 3403 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) a ) )  C_  ( X  X.  X
) )
76ad2antrr 725 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  a  e.  RR+ )  ->  ( `' D " ( 0 [,) a ) ) 
C_  ( X  X.  X ) )
8 cnvexg 6523 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
9 imaexg 6514 . . . . . . . . 9  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
10 elpwg 3867 . . . . . . . . 9  |-  ( ( `' D " ( 0 [,) a ) )  e.  _V  ->  (
( `' D "
( 0 [,) a
) )  e.  ~P ( X  X.  X
)  <->  ( `' D " ( 0 [,) a
) )  C_  ( X  X.  X ) ) )
118, 9, 103syl 20 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  ( ( `' D " ( 0 [,) a ) )  e.  ~P ( X  X.  X )  <->  ( `' D " ( 0 [,) a ) )  C_  ( X  X.  X
) ) )
1211ad2antrr 725 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  a  e.  RR+ )  ->  (
( `' D "
( 0 [,) a
) )  e.  ~P ( X  X.  X
)  <->  ( `' D " ( 0 [,) a
) )  C_  ( X  X.  X ) ) )
137, 12mpbird 232 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  a  e.  RR+ )  ->  ( `' D " ( 0 [,) a ) )  e.  ~P ( X  X.  X ) )
1413ralrimiva 2798 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A. a  e.  RR+  ( `' D " ( 0 [,) a
) )  e.  ~P ( X  X.  X
) )
15 eqid 2442 . . . . . 6  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
1615rnmptss 5871 . . . . 5  |-  ( A. a  e.  RR+  ( `' D " ( 0 [,) a ) )  e.  ~P ( X  X.  X )  ->  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  C_  ~P ( X  X.  X ) )
1714, 16syl 16 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  C_  ~P ( X  X.  X ) )
181, 17syl5eqss 3399 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  F  C_ 
~P ( X  X.  X ) )
19 simpr 461 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  e.  F )
2018, 19sseldd 3356 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  e.  ~P ( X  X.  X ) )
2120elpwid 3869 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2714   _Vcvv 2971    C_ wss 3327   ~Pcpw 3859    e. cmpt 4349    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842   -->wf 5413   ` cfv 5417  (class class class)co 6090   0cc0 9281   RR*cxr 9416   RR+crp 10990   [,)cico 11301  PsMetcpsmet 17799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  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 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7215  df-xr 9421  df-psmet 17808
This theorem is referenced by:  metustrel  20130  metustsym  20136
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