MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metustss Unicode version

Theorem metustss 18537
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 5183 . . . . . . . . 9  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
3 psmetf 18290 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
4 fdm 5554 . . . . . . . . . 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 3356 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) a ) )  C_  ( X  X.  X
) )
76ad2antrr 707 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  a  e.  RR+ )  ->  ( `' D " ( 0 [,) a ) ) 
C_  ( X  X.  X ) )
8 cnvexg 5364 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
9 imaexg 5176 . . . . . . . . 9  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
10 elpwg 3766 . . . . . . . . 9  |-  ( ( `' D " ( 0 [,) a ) )  e.  _V  ->  (
( `' D "
( 0 [,) a
) )  e.  ~P ( X  X.  X
)  <->  ( `' D " ( 0 [,) a
) )  C_  ( X  X.  X ) ) )
118, 9, 103syl 19 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  ( ( `' D " ( 0 [,) a ) )  e.  ~P ( X  X.  X )  <->  ( `' D " ( 0 [,) a ) )  C_  ( X  X.  X
) ) )
1211ad2antrr 707 . . . . . . 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 224 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  a  e.  RR+ )  ->  ( `' D " ( 0 [,) a ) )  e.  ~P ( X  X.  X ) )
1413ralrimiva 2749 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A. a  e.  RR+  ( `' D " ( 0 [,) a
) )  e.  ~P ( X  X.  X
) )
15 eqid 2404 . . . . . 6  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
1615rnmptss 5856 . . . . 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 3352 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  F  C_ 
~P ( X  X.  X ) )
19 simpr 448 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  e.  F )
2018, 19sseldd 3309 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  e.  ~P ( X  X.  X ) )
2120elpwid 3768 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946   RR*cxr 9075   RR+crp 10568   [,)cico 10874  PsMetcpsmet 16640
This theorem is referenced by:  metustrel  18539  metustsym  18545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-xr 9080  df-psmet 16649
  Copyright terms: Public domain W3C validator