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Theorem metustssOLD 20103
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 5184 . . . . . . . . 9  |-  ( `' D " ( 0 [,) a ) ) 
C_  dom  D
3 xmetf 19879 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
4 fdm 5558 . . . . . . . . . 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 3399 . . . . . . . 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 6519 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
9 imaexg 6510 . . . . . . . . 9  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
10 elpwg 3863 . . . . . . . . 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 2794 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A. a  e.  RR+  ( `' D " ( 0 [,) a
) )  e.  ~P ( X  X.  X
) )
15 eqid 2438 . . . . . 6  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
1615rnmptss 5867 . . . . 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 3395 . . 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 3352 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  e.  ~P ( X  X.  X
) )
2120elpwid 3865 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 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967    C_ wss 3323   ~Pcpw 3855    e. cmpt 4345    X. cxp 4833   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838   -->wf 5409   ` cfv 5413  (class class class)co 6086   0cc0 9274   RR*cxr 9409   RR+crp 10983   [,)cico 11294   *Metcxmt 17776
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-xr 9414  df-xmet 17785
This theorem is referenced by:  metustrelOLD  20105  metustsymOLD  20111
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