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Theorem metustrelOLD 20788
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 20786 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  C_  ( X  X.  X ) )
3 xpss 5102 . . 3  |-  ( X  X.  X )  C_  ( _V  X.  _V )
42, 3syl6ss 3511 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  A  C_  ( _V  X.  _V ) )
5 df-rel 5001 . 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 369    = wceq 1374    e. wcel 1762   _Vcvv 3108    C_ wss 3471    |-> cmpt 4500    X. cxp 4992   `'ccnv 4993   ran crn 4995   "cima 4997   Rel wrel 4999   ` cfv 5581  (class class class)co 6277   0cc0 9483   RR+crp 11211   [,)cico 11522   *Metcxmt 18169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-xr 9623  df-xmet 18178
This theorem is referenced by: (None)
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