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Theorem unirnbl 17969
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.)
Assertion
Ref Expression
unirnbl  |-  ( D  e.  ( * Met `  X )  ->  U. ran  ( ball `  D )  =  X )

Proof of Theorem unirnbl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 blf 17961 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
2 frn 5395 . . . 4  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  ->  ran  ( ball `  D
)  C_  ~P X
)
31, 2syl 15 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ran  ( ball `  D )  C_ 
~P X )
4 sspwuni 3987 . . 3  |-  ( ran  ( ball `  D
)  C_  ~P X  <->  U.
ran  ( ball `  D
)  C_  X )
53, 4sylib 188 . 2  |-  ( D  e.  ( * Met `  X )  ->  U. ran  ( ball `  D )  C_  X )
6 1rp 10358 . . . . . 6  |-  1  e.  RR+
7 blcntr 17964 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  X  /\  1  e.  RR+ )  ->  x  e.  ( x ( ball `  D
) 1 ) )
86, 7mp3an3 1266 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  X
)  ->  x  e.  ( x ( ball `  D ) 1 ) )
9 rpxr 10361 . . . . . . 7  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
106, 9ax-mp 8 . . . . . 6  |-  1  e.  RR*
11 blelrn 17967 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  X  /\  1  e.  RR* )  ->  ( x ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )
1210, 11mp3an3 1266 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  X
)  ->  ( x
( ball `  D )
1 )  e.  ran  ( ball `  D )
)
13 elunii 3832 . . . . 5  |-  ( ( x  e.  ( x ( ball `  D
) 1 )  /\  ( x ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )  ->  x  e.  U. ran  ( ball `  D ) )
148, 12, 13syl2anc 642 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  x  e.  X
)  ->  x  e.  U.
ran  ( ball `  D
) )
1514ex 423 . . 3  |-  ( D  e.  ( * Met `  X )  ->  (
x  e.  X  ->  x  e.  U. ran  ( ball `  D ) ) )
1615ssrdv 3185 . 2  |-  ( D  e.  ( * Met `  X )  ->  X  C_ 
U. ran  ( ball `  D ) )
175, 16eqssd 3196 1  |-  ( D  e.  ( * Met `  X )  ->  U. ran  ( ball `  D )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   1c1 8738   RR*cxr 8866   RR+crp 10354   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  blbas  17976  mopntopon  17985  elmopn  17988  imasf1oxms  18035  metss  18054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-rp 10355  df-xmet 16373  df-bl 16375
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