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Theorem blrn 20119
Description: Membership in the range of the ball function. Note that  ran  ( ball `  D ) is the collection of all balls for metric 
D. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
blrn  |-  ( D  e.  ( *Met `  X )  ->  ( A  e.  ran  ( ball `  D )  <->  E. x  e.  X  E. r  e.  RR*  A  =  ( x ( ball `  D
) r ) ) )
Distinct variable groups:    x, r, A    D, r, x    X, r, x

Proof of Theorem blrn
StepHypRef Expression
1 blf 20117 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
2 ffn 5670 . 2  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  -> 
( ball `  D )  Fn  ( X  X.  RR* ) )
3 ovelrn 6352 . 2  |-  ( (
ball `  D )  Fn  ( X  X.  RR* )  ->  ( A  e. 
ran  ( ball `  D
)  <->  E. x  e.  X  E. r  e.  RR*  A  =  ( x (
ball `  D )
r ) ) )
41, 2, 33syl 20 1  |-  ( D  e.  ( *Met `  X )  ->  ( A  e.  ran  ( ball `  D )  <->  E. x  e.  X  E. r  e.  RR*  A  =  ( x ( ball `  D
) r ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   E.wrex 2800   ~Pcpw 3971    X. cxp 4949   ran crn 4952    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   RR*cxr 9531   *Metcxmt 17929   ballcbl 17931
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-xr 9536  df-psmet 17937  df-xmet 17938  df-bl 17940
This theorem is referenced by:  blss  20135  imasf1oxms  20199  prdsxmslem2  20239  blssioo  20507
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