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Theorem metustssOLD 21350
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 5179 . . . . . . . . 9  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
3 xmetf 21126 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
4 fdm 5720 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
53, 4syl 17 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  dom  D  =  ( X  X.  X ) )
62, 5syl5sseq 3492 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " ( 0 [,) a ) ) 
C_  ( X  X.  X ) )
76ad2antrr 726 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  a  e.  RR+ )  ->  ( `' D " ( 0 [,) a ) ) 
C_  ( X  X.  X ) )
8 cnvexg 6732 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
9 imaexg 6723 . . . . . . . . 9  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
10 elpwg 3965 . . . . . . . . 9  |-  ( ( `' D " ( 0 [,) a ) )  e.  _V  ->  (
( `' D "
( 0 [,) a
) )  e.  ~P ( X  X.  X
)  <->  ( `' D " ( 0 [,) a
) )  C_  ( X  X.  X ) ) )
118, 9, 103syl 18 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  (
( `' D "
( 0 [,) a
) )  e.  ~P ( X  X.  X
)  <->  ( `' D " ( 0 [,) a
) )  C_  ( X  X.  X ) ) )
1211ad2antrr 726 . . . . . . 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 234 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  a  e.  RR+ )  ->  ( `' D " ( 0 [,) a ) )  e.  ~P ( X  X.  X ) )
1413ralrimiva 2820 . . . . 5  |-  ( ( D  e.  ( *Met `  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 6041 . . . . 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 17 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  C_  ~P ( X  X.  X ) )
181, 17syl5eqss 3488 . . 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 3445 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  e.  ~P ( X  X.  X
) )
2120elpwid 3967 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  C_  ( X  X.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   _Vcvv 3061    C_ wss 3416   ~Pcpw 3957    |-> cmpt 4455    X. cxp 4823   `'ccnv 4824   dom cdm 4825   ran crn 4826   "cima 4828   -->wf 5567   ` cfv 5571  (class class class)co 6280   0cc0 9524   RR*cxr 9659   RR+crp 11267   [,)cico 11586   *Metcxmt 18725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-map 7461  df-xr 9664  df-xmet 18734
This theorem is referenced by:  metustrelOLD  21352  metustsymOLD  21358
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