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Theorem metustrelOLD 21350
Description: Elements of the filter base generated by the metric  D are relations. (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
metustrelOLD  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  Rel  A )
Distinct variable groups:    D, a    X, a    A, a    F, a

Proof of Theorem metustrelOLD
StepHypRef Expression
1 metust.1 . . . 4  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
21metustssOLD 21348 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  C_  ( X  X.  X ) )
3 xpss 4930 . . 3  |-  ( X  X.  X )  C_  ( _V  X.  _V )
42, 3syl6ss 3454 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  C_  ( _V  X.  _V ) )
5 df-rel 4830 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
64, 5sylibr 212 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  Rel  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    C_ wss 3414    |-> cmpt 4453    X. cxp 4821   `'ccnv 4822   ran crn 4824   "cima 4826   Rel wrel 4828   ` cfv 5569  (class class class)co 6278   0cc0 9522   RR+crp 11265   [,)cico 11584   *Metcxmt 18723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-xr 9662  df-xmet 18732
This theorem is referenced by: (None)
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