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Theorem blval 17948
Description: The ball around a point  P is the set of all points whose distance from  P is less than the ball's radius  R. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Assertion
Ref Expression
blval  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  =  { x  e.  X  |  ( P D x )  < 
R } )
Distinct variable groups:    x, P    x, D    x, R    x, X

Proof of Theorem blval
Dummy variables  r 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blfval 17947 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D )  =  ( y  e.  X ,  r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
213ad2ant1 976 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ball `  D
)  =  ( y  e.  X ,  r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
3 simprl 732 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
y  =  P )
43oveq1d 5873 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( y D x )  =  ( P D x ) )
5 simprr 733 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
r  =  R )
64, 5breq12d 4036 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( ( y D x )  <  r  <->  ( P D x )  <  R ) )
76rabbidv 2780 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  ->  { x  e.  X  |  ( y D x )  <  r }  =  { x  e.  X  |  ( P D x )  < 
R } )
8 simp2 956 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  P  e.  X )
9 simp3 957 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  R  e.  RR* )
10 elfvdm 5554 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  X  e.  dom  * Met )
11103ad2ant1 976 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  X  e.  dom  * Met )
12 rabexg 4164 . . 3  |-  ( X  e.  dom  * Met  ->  { x  e.  X  |  ( P D x )  <  R }  e.  _V )
1311, 12syl 15 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  { x  e.  X  |  ( P D x )  <  R }  e.  _V )
142, 7, 8, 9, 13ovmpt2d 5975 1  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  =  { x  e.  X  |  ( P D x )  < 
R } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   class class class wbr 4023   dom cdm 4689   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   RR*cxr 8866    < clt 8867   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  elbl  17949  metss2lem  18057  stdbdbl  18063  nmhmcn  18601  isbnd3  26508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-xr 8871  df-xmet 16373  df-bl 16375
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