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Theorem unirnblps 12628
Description: The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
unirnblps  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  =  X )

Proof of Theorem unirnblps
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 blfps 12615 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
21frnd 5289 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ran  ( ball `  D )  C_  ~P X )
3 sspwuni 3904 . . 3  |-  ( ran  ( ball `  D
)  C_  ~P X  <->  U.
ran  ( ball `  D
)  C_  X )
42, 3sylib 121 . 2  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  C_  X )
5 1rp 9473 . . . 4  |-  1  e.  RR+
6 blcntrps 12621 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  1  e.  RR+ )  ->  x  e.  ( x ( ball `  D ) 1 ) )
75, 6mp3an3 1305 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  x  e.  ( x ( ball `  D ) 1 ) )
8 rpxr 9477 . . . . 5  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
95, 8ax-mp 5 . . . 4  |-  1  e.  RR*
10 blelrnps 12625 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  1  e.  RR* )  ->  (
x ( ball `  D
) 1 )  e. 
ran  ( ball `  D
) )
119, 10mp3an3 1305 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x ( ball `  D
) 1 )  e. 
ran  ( ball `  D
) )
12 elunii 3748 . . 3  |-  ( ( x  e.  ( x ( ball `  D
) 1 )  /\  ( x ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )  ->  x  e.  U. ran  ( ball `  D ) )
137, 11, 12syl2anc 409 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  x  e.  U. ran  ( ball `  D ) )
144, 13eqelssd 3120 1  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481    C_ wss 3075   ~Pcpw 3514   U.cuni 3743    X. cxp 4544   ran crn 4547   ` cfv 5130  (class class class)co 5781   1c1 7644   RR*cxr 7822   RR+crp 9469  PsMetcpsmet 12185   ballcbl 12188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1re 7737  ax-addrcl 7740  ax-0lt1 7749  ax-rnegex 7752
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-rp 9470  df-psmet 12193  df-bl 12196
This theorem is referenced by: (None)
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