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Theorem metustssOLD 20791
Description: Range of the elements of the filter base generated by the metric  D. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metustssOLD  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  C_  ( X  X.  X ) )
Distinct variable groups:    D, a    X, a    A, a    F, a

Proof of Theorem metustssOLD
StepHypRef Expression
1 metust.1 . . . 4  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
2 cnvimass 5355 . . . . . . . . 9  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
3 xmetf 20567 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
4 fdm 5733 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
53, 4syl 16 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  dom  D  =  ( X  X.  X ) )
62, 5syl5sseq 3552 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " ( 0 [,) a ) ) 
C_  ( X  X.  X ) )
76ad2antrr 725 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  a  e.  RR+ )  ->  ( `' D " ( 0 [,) a ) ) 
C_  ( X  X.  X ) )
8 cnvexg 6727 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
9 imaexg 6718 . . . . . . . . 9  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
10 elpwg 4018 . . . . . . . . 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.  ( *Met `  X )  ->  (
( `' D "
( 0 [,) a
) )  e.  ~P ( X  X.  X
)  <->  ( `' D " ( 0 [,) a
) )  C_  ( X  X.  X ) ) )
1211ad2antrr 725 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  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.  ( *Met `  X
)  /\  A  e.  F )  /\  a  e.  RR+ )  ->  ( `' D " ( 0 [,) a ) )  e.  ~P ( X  X.  X ) )
1413ralrimiva 2878 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A. a  e.  RR+  ( `' D " ( 0 [,) a
) )  e.  ~P ( X  X.  X
) )
15 eqid 2467 . . . . . 6  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
1615rnmptss 6048 . . . . 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.  ( *Met `  X )  /\  A  e.  F
)  ->  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  C_  ~P ( X  X.  X ) )
181, 17syl5eqss 3548 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  F  C_  ~P ( X  X.  X
) )
19 simpr 461 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  e.  F )
2018, 19sseldd 3505 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  e.  ~P ( X  X.  X
) )
2120elpwid 4020 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  C_  ( X  X.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   -->wf 5582   ` cfv 5586  (class class class)co 6282   0cc0 9488   RR*cxr 9623   RR+crp 11216   [,)cico 11527   *Metcxmt 18174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-xr 9628  df-xmet 18183
This theorem is referenced by:  metustrelOLD  20793  metustsymOLD  20799
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