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Theorem unirnblps 21214
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 21201 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
2 frn 5720 . . . 4  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  ->  ran  ( ball `  D
)  C_  ~P X
)
31, 2syl 17 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ran  ( ball `  D )  C_  ~P X )
4 sspwuni 4360 . . 3  |-  ( ran  ( ball `  D
)  C_  ~P X  <->  U.
ran  ( ball `  D
)  C_  X )
53, 4sylib 196 . 2  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  C_  X )
6 1rp 11269 . . . . . 6  |-  1  e.  RR+
7 blcntrps 21207 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  1  e.  RR+ )  ->  x  e.  ( x ( ball `  D ) 1 ) )
86, 7mp3an3 1315 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  x  e.  ( x ( ball `  D ) 1 ) )
9 rpxr 11272 . . . . . . 7  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
106, 9ax-mp 5 . . . . . 6  |-  1  e.  RR*
11 blelrnps 21211 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  1  e.  RR* )  ->  (
x ( ball `  D
) 1 )  e. 
ran  ( ball `  D
) )
1210, 11mp3an3 1315 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x ( ball `  D
) 1 )  e. 
ran  ( ball `  D
) )
13 elunii 4196 . . . . 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.  (PsMet `  X )  /\  x  e.  X )  ->  x  e.  U. ran  ( ball `  D ) )
1514ex 432 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X  ->  x  e. 
U. ran  ( ball `  D ) ) )
1615ssrdv 3448 . 2  |-  ( D  e.  (PsMet `  X
)  ->  X  C_  U. ran  ( ball `  D )
)
175, 16eqssd 3459 1  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3414   ~Pcpw 3955   U.cuni 4191    X. cxp 4821   ran crn 4824   -->wf 5565   ` cfv 5569  (class class class)co 6278   1c1 9523   RR*cxr 9657   RR+crp 11265  PsMetcpsmet 18722   ballcbl 18725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-rp 11266  df-psmet 18731  df-bl 18734
This theorem is referenced by:  psmetutop  21378
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