Theorem blval 22191

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 22189	. . 3 |- ( D e. ( *Met ` X ) -> ( ball ` D ) = ( y e. X , r e. RR* |-> { x e. X | ( y D x ) < r } ) )
2	1	3ad2ant1 1082	. 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 794	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> y = P )
4	3	oveq1d 6665	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> ( y D x ) = ( P D x ) )
5		simprr 796	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ ( y = P /\ r = R ) ) -> r = R )
6	4, 5	breq12d 4666	. . 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 3189	. 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 1062	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> P e. X )
9		simp3 1063	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> R e. RR* )
10		elfvdm 6220	. . . 4 |- ( D e. ( *Met ` X ) -> X e. dom *Met )
11	10	3ad2ant1 1082	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> X e. dom *Met )
12		rabexg 4812	. . 3 |- ( X e. dom *Met -> { x e. X | ( P D x ) < R } e. _V )
13	11, 12	syl 17	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> { x e. X | ( P D x ) < R } e. _V )
14	2, 7, 8, 9, 13	ovmpt2d 6788	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RR*cxr 10073   < clt 10074   *Metcxmt 19731   ballcbl 19733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-xr 10078  df-psmet 19738  df-xmet 19739  df-bl 19741

This theorem is referenced by:  elbl  22193  metss2lem  22316  stdbdbl  22322  nmhmcn  22920  lgamucov  24764  isbnd3  33583