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Theorem blfvalps 12591
Description: The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
blfvalps  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) )
Distinct variable groups:    x, r, y, D    X, r, x, y

Proof of Theorem blfvalps
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 df-bl 12196 . . 3  |-  ball  =  ( d  e.  _V  |->  ( x  e.  dom  dom  d ,  r  e. 
RR*  |->  { y  e. 
dom  dom  d  |  ( x d y )  <  r } ) )
21a1i 9 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ball  =  ( d  e.  _V  |->  ( x  e.  dom  dom  d ,  r  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  < 
r } ) ) )
3 dmeq 4746 . . . . 5  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 4748 . . . 4  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 psmetdmdm 12530 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
65eqcomd 2146 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  dom  dom  D  =  X )
74, 6sylan9eqr 2195 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
8 eqidd 2141 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  RR*  =  RR* )
9 simpr 109 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  d  =  D )
109oveqd 5798 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x d y )  =  ( x D y ) )
1110breq1d 3946 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
( x d y )  <  r  <->  ( x D y )  < 
r ) )
127, 11rabeqbidv 2684 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  { y  e.  dom  dom  d  |  ( x d y )  <  r }  =  { y  e.  X  |  (
x D y )  <  r } )
137, 8, 12mpoeq123dv 5840 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x  e.  dom  dom  d ,  r  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  < 
r } )  =  ( x  e.  X ,  r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) )
14 elex 2700 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  _V )
15 ssrab2 3186 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
16 psmetrel 12528 . . . . . . . . 9  |-  Rel PsMet
17 relelfvdm 5460 . . . . . . . . 9  |-  ( ( Rel PsMet  /\  D  e.  (PsMet `  X ) )  ->  X  e.  dom PsMet )
1816, 17mpan 421 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
1918adantr 274 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  r  e.  RR* ) )  ->  X  e.  dom PsMet )
20 elpw2g 4088 . . . . . . 7  |-  ( X  e.  dom PsMet  ->  ( { y  e.  X  | 
( x D y )  <  r }  e.  ~P X  <->  { y  e.  X  |  (
x D y )  <  r }  C_  X ) )
2119, 20syl 14 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  r  e.  RR* ) )  ->  ( { y  e.  X  |  ( x D y )  <  r }  e.  ~P X  <->  { y  e.  X  |  ( x D y )  <  r }  C_  X ) )
2215, 21mpbiri 167 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  r  e.  RR* ) )  ->  { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X )
2322ralrimivva 2517 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  X  A. r  e.  RR*  { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X )
24 eqid 2140 . . . . 5  |-  ( x  e.  X ,  r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } )  =  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } )
2524fmpo 6106 . . . 4  |-  ( A. x  e.  X  A. r  e.  RR*  { y  e.  X  |  ( x D y )  <  r }  e.  ~P X  <->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } ) : ( X  X.  RR* )
--> ~P X )
2623, 25sylib 121 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } ) : ( X  X.  RR* )
--> ~P X )
27 xrex 9668 . . . 4  |-  RR*  e.  _V
28 xpexg 4660 . . . 4  |-  ( ( X  e.  dom PsMet  /\  RR*  e.  _V )  ->  ( X  X.  RR* )  e.  _V )
2918, 27, 28sylancl 410 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( X  X.  RR* )  e.  _V )
3018pwexd 4112 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ~P X  e.  _V )
31 fex2 5298 . . 3  |-  ( ( ( x  e.  X ,  r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) : ( X  X.  RR* ) --> ~P X  /\  ( X  X.  RR* )  e.  _V  /\  ~P X  e.  _V )  ->  (
x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } )  e.  _V )
3226, 29, 30, 31syl3anc 1217 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } )  e. 
_V )
332, 13, 14, 32fvmptd 5509 1  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417   {crab 2421   _Vcvv 2689    C_ wss 3075   ~Pcpw 3514   class class class wbr 3936    |-> cmpt 3996    X. cxp 4544   dom cdm 4546   Rel wrel 4551   -->wf 5126   ` cfv 5130  (class class class)co 5781    e. cmpo 5783   RR*cxr 7822    < clt 7823  PsMetcpsmet 12185   ballcbl 12188
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 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735
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 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-pnf 7825  df-mnf 7826  df-xr 7827  df-psmet 12193  df-bl 12196
This theorem is referenced by:  blfval  12592  blvalps  12594  blfps  12615
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