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Theorem unirnbl 20097
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 20084 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
2 frn 5649 . . . 4  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  ->  ran  ( ball `  D
)  C_  ~P X
)
31, 2syl 16 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ran  ( ball `  D )  C_ 
~P X )
4 sspwuni 4340 . . 3  |-  ( ran  ( ball `  D
)  C_  ~P X  <->  U.
ran  ( ball `  D
)  C_  X )
53, 4sylib 196 . 2  |-  ( D  e.  ( *Met `  X )  ->  U. ran  ( ball `  D )  C_  X )
6 1rp 11082 . . . . . 6  |-  1  e.  RR+
7 blcntr 20090 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  1  e.  RR+ )  ->  x  e.  ( x ( ball `  D
) 1 ) )
86, 7mp3an3 1304 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  x  e.  ( x ( ball `  D ) 1 ) )
9 rpxr 11085 . . . . . . 7  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
106, 9ax-mp 5 . . . . . 6  |-  1  e.  RR*
11 blelrn 20094 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  1  e.  RR* )  ->  ( x ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )
1210, 11mp3an3 1304 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x
( ball `  D )
1 )  e.  ran  ( ball `  D )
)
13 elunii 4180 . . . . 5  |-  ( ( x  e.  ( x ( ball `  D
) 1 )  /\  ( x ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )  ->  x  e.  U. ran  ( ball `  D ) )
148, 12, 13syl2anc 661 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  x  e.  U.
ran  ( ball `  D
) )
1514ex 434 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  ->  x  e.  U. ran  ( ball `  D ) ) )
1615ssrdv 3446 . 2  |-  ( D  e.  ( *Met `  X )  ->  X  C_ 
U. ran  ( ball `  D ) )
175, 16eqssd 3457 1  |-  ( D  e.  ( *Met `  X )  ->  U. ran  ( ball `  D )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757    C_ wss 3412   ~Pcpw 3944   U.cuni 4175    X. cxp 4922   ran crn 4925   -->wf 5498   ` cfv 5502  (class class class)co 6176   1c1 9370   RR*cxr 9504   RR+crp 11078   *Metcxmt 17896   ballcbl 17898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-po 4725  df-so 4726  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-er 7187  df-map 7302  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-rp 11079  df-psmet 17904  df-xmet 17905  df-bl 17907
This theorem is referenced by:  blbas  20107  mopntopon  20116  elmopn  20119  imasf1oxms  20166  metss  20185  metutopOLD  20259
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