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Theorem blrnps 21201
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.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blrnps  |-  ( D  e.  (PsMet `  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 blrnps
StepHypRef Expression
1 blfps 21199 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
2 ffn 5713 . 2  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  -> 
( ball `  D )  Fn  ( X  X.  RR* ) )
3 ovelrn 6431 . 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.  (PsMet `  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 1405    e. wcel 1842   E.wrex 2754   ~Pcpw 3954    X. cxp 4820   ran crn 4823    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277   RR*cxr 9656  PsMetcpsmet 18720   ballcbl 18723
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 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-map 7458  df-xr 9661  df-psmet 18729  df-bl 18732
This theorem is referenced by:  blssps  21217
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