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Theorem unirnblps 19992
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 19979 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
2 frn 5563 . . . 4  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  ->  ran  ( ball `  D
)  C_  ~P X
)
31, 2syl 16 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ran  ( ball `  D )  C_  ~P X )
4 sspwuni 4254 . . 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 10993 . . . . . 6  |-  1  e.  RR+
7 blcntrps 19985 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  1  e.  RR+ )  ->  x  e.  ( x ( ball `  D ) 1 ) )
86, 7mp3an3 1303 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  x  e.  ( x ( ball `  D ) 1 ) )
9 rpxr 10996 . . . . . . 7  |-  ( 1  e.  RR+  ->  1  e. 
RR* )
106, 9ax-mp 5 . . . . . 6  |-  1  e.  RR*
11 blelrnps 19989 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  1  e.  RR* )  ->  (
x ( ball `  D
) 1 )  e. 
ran  ( ball `  D
) )
1210, 11mp3an3 1303 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  ->  (
x ( ball `  D
) 1 )  e. 
ran  ( ball `  D
) )
13 elunii 4094 . . . . 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.  (PsMet `  X )  /\  x  e.  X )  ->  x  e.  U. ran  ( ball `  D ) )
1514ex 434 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X  ->  x  e. 
U. ran  ( ball `  D ) ) )
1615ssrdv 3360 . 2  |-  ( D  e.  (PsMet `  X
)  ->  X  C_  U. ran  ( ball `  D )
)
175, 16eqssd 3371 1  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3326   ~Pcpw 3858   U.cuni 4089    X. cxp 4836   ran crn 4839   -->wf 5412   ` cfv 5416  (class class class)co 6089   1c1 9281   RR*cxr 9415   RR+crp 10989  PsMetcpsmet 17798   ballcbl 17801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-rp 10990  df-psmet 17807  df-bl 17810
This theorem is referenced by:  psmetutop  20156
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