Theorem blval 12597

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.

Step	Hyp	Ref	Expression
1		blfval 12594	. . 3 |- ( D e. ( *Met ` X ) -> ( ball ` D ) = ( y e. X , r e. RR* |-> { x e. X | ( y D x ) < r } ) )
2	1	3ad2ant1 1003	. 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 521	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> y = P )
4	3	oveq1d 5797	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> ( y D x ) = ( P D x ) )
5		simprr 522	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> r = R )
6	4, 5	breq12d 3950	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> ( ( y D x ) < r <-> ( P D x ) < R ) )
7	6	rabbidv 2678	. 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 983	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> P e. X )
9		simp3 984	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> R e. RR* )
10		xmetrel 12551	. . . . 5 |- Rel *Met
11		relelfvdm 5461	. . . . 5 |- ( ( Rel *Met /\ D e. ( *Met ` X ) ) -> X e. dom *Met )
12	10, 11	mpan 421	. . . 4 |- ( D e. ( *Met ` X ) -> X e. dom *Met )
13	12	3ad2ant1 1003	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> X e. dom *Met )
14		rabexg 4079	. . 3 |- ( X e. dom *Met -> { x e. X | ( P D x ) < R } e. _V )
15	13, 14	syl 14	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> { x e. X | ( P D x ) < R } e. _V )
16	2, 7, 8, 9, 15	ovmpod 5906	1 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) = { x e. X | ( P D x ) < R } )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1332   e. wcel 1481   {crab 2421   _Vcvv 2689   class class class wbr 3937   dom cdm 4547   Rel wrel 4552   ` cfv 5131  (class class class)co 5782   e. cmpo 5784   RR*cxr 7823   < clt 7824   *Metcxmet 12188   ballcbl 12190

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-pnf 7826  df-mnf 7827  df-xr 7828  df-psmet 12195  df-xmet 12196  df-bl 12198

This theorem is referenced by:  elbl  12599  metss2lem  12705  bdbl  12711  xmetxpbl  12716