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Theorem blval 12568
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 12565 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D )  =  ( y  e.  X ,  r  e.  RR*  |->  { x  e.  X  |  (
y D x )  <  r } ) )
213ad2ant1 1002 . 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 520 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
y  =  P )
43oveq1d 5789 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( y D x )  =  ( P D x ) )
5 simprr 521 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
r  =  R )
64, 5breq12d 3942 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( y  =  P  /\  r  =  R ) )  -> 
( ( y D x )  <  r  <->  ( P D x )  <  R ) )
76rabbidv 2675 . 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 982 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  P  e.  X )
9 simp3 983 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  R  e.  RR* )
10 xmetrel 12522 . . . . 5  |-  Rel  *Met
11 relelfvdm 5453 . . . . 5  |-  ( ( Rel  *Met  /\  D  e.  ( *Met `  X ) )  ->  X  e.  dom  *Met )
1210, 11mpan 420 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
13123ad2ant1 1002 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  X  e.  dom  *Met )
14 rabexg 4071 . . 3  |-  ( X  e.  dom  *Met  ->  { x  e.  X  |  ( P D x )  <  R }  e.  _V )
1513, 14syl 14 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  { x  e.  X  |  ( P D x )  <  R }  e.  _V )
162, 7, 8, 9, 15ovmpod 5898 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480   {crab 2420   _Vcvv 2686   class class class wbr 3929   dom cdm 4539   Rel wrel 4544   ` cfv 5123  (class class class)co 5774    e. cmpo 5776   RR*cxr 7806    < clt 7807   *Metcxmet 12159   ballcbl 12161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-pnf 7809  df-mnf 7810  df-xr 7811  df-psmet 12166  df-xmet 12167  df-bl 12169
This theorem is referenced by:  elbl  12570  metss2lem  12676  bdbl  12682  xmetxpbl  12687
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