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Theorem unirnbl 21089
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 21076 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
2 frn 5719 . . . 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 4404 . . 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 11225 . . . . . 6  |-  1  e.  RR+
7 blcntr 21082 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  1  e.  RR+ )  ->  x  e.  ( x ( ball `  D
) 1 ) )
86, 7mp3an3 1311 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  x  e.  ( x ( ball `  D ) 1 ) )
9 rpxr 11228 . . . . . . 7  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
106, 9ax-mp 5 . . . . . 6  |-  1  e.  RR*
11 blelrn 21086 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  1  e.  RR* )  ->  ( x ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )
1210, 11mp3an3 1311 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( x
( ball `  D )
1 )  e.  ran  ( ball `  D )
)
13 elunii 4240 . . . . 5  |-  ( ( x  e.  ( x ( ball `  D
) 1 )  /\  ( x ( ball `  D ) 1 )  e.  ran  ( ball `  D ) )  ->  x  e.  U. ran  ( ball `  D ) )
148, 12, 13syl2anc 659 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  x  e.  U.
ran  ( ball `  D
) )
1514ex 432 . . 3  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X  ->  x  e.  U. ran  ( ball `  D ) ) )
1615ssrdv 3495 . 2  |-  ( D  e.  ( *Met `  X )  ->  X  C_ 
U. ran  ( ball `  D ) )
175, 16eqssd 3506 1  |-  ( D  e.  ( *Met `  X )  ->  U. ran  ( ball `  D )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   ~Pcpw 3999   U.cuni 4235    X. cxp 4986   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   1c1 9482   RR*cxr 9616   RR+crp 11221   *Metcxmt 18598   ballcbl 18600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-rp 11222  df-psmet 18606  df-xmet 18607  df-bl 18609
This theorem is referenced by:  blbas  21099  mopntopon  21108  elmopn  21111  imasf1oxms  21158  metss  21177  metutopOLD  21251
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