MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  blrn Structured version   Unicode version

Theorem blrn 21094
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 21092 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
2 ffn 5668 . 2  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  -> 
( ball `  D )  Fn  ( X  X.  RR* ) )
3 ovelrn 6386 . 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 1403    e. wcel 1840   E.wrex 2752   ~Pcpw 3952    X. cxp 4938   ran crn 4941    Fn wfn 5518   -->wf 5519   ` cfv 5523  (class class class)co 6232   RR*cxr 9575   *Metcxmt 18613   ballcbl 18615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-map 7377  df-xr 9580  df-psmet 18621  df-xmet 18622  df-bl 18624
This theorem is referenced by:  blss  21110  imasf1oxms  21174  prdsxmslem2  21214  blssioo  21482
  Copyright terms: Public domain W3C validator