MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  blfval Unicode version

Theorem blfval 17943
Description: The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Assertion
Ref Expression
blfval  |-  ( D  e.  ( * Met `  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 blfval
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 df-bl 16371 . . 3  |-  ball  =  ( d  e.  U. ran  * Met  |->  ( x  e.  dom  dom  d ,  r  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  <  r } ) )
21a1i 10 . 2  |-  ( D  e.  ( * Met `  X )  ->  ball  =  ( d  e.  U. ran  * Met  |->  ( x  e.  dom  dom  d ,  r  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  <  r } ) ) )
3 dmeq 4878 . . . . 5  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 4880 . . . 4  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 xmetdmdm 17896 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  X  =  dom  dom  D )
65eqcomd 2289 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  X )
74, 6sylan9eqr 2338 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
8 eqidd 2285 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  RR*  =  RR* )
9 simpr 447 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  d  =  D )
109oveqd 5837 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( x
d y )  =  ( x D y ) )
1110breq1d 4034 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( (
x d y )  <  r  <->  ( x D y )  < 
r ) )
127, 11rabeqbidv 2784 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  { y  e.  dom  dom  d  | 
( x d y )  <  r }  =  { y  e.  X  |  ( x D y )  < 
r } )
137, 8, 12mpt2eq123dv 5872 . 2  |-  ( ( D  e.  ( * Met `  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 fvssunirn 5513 . . 3  |-  ( * Met `  X ) 
C_  U. ran  * Met
1514sseli 3177 . 2  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
16 ssrab2 3259 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
17 elfvdm 5516 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  X  e.  dom  * Met )
1817adantr 451 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  ( x  e.  X  /\  r  e. 
RR* ) )  ->  X  e.  dom  * Met )
19 elpw2g 4168 . . . . . . 7  |-  ( X  e.  dom  * Met  ->  ( { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X 
<->  { y  e.  X  |  ( x D y )  <  r }  C_  X ) )
2018, 19syl 15 . . . . . 6  |-  ( ( D  e.  ( * Met `  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 ) )
2116, 20mpbiri 224 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  ( x  e.  X  /\  r  e. 
RR* ) )  ->  { y  e.  X  |  ( x D y )  <  r }  e.  ~P X
)
2221ralrimivva 2636 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  A. x  e.  X  A. r  e.  RR*  { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X )
23 eqid 2284 . . . . 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 } )
2423fmpt2 6153 . . . 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 )
2522, 24sylib 188 . . 3  |-  ( D  e.  ( * Met `  X )  ->  (
x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) : ( X  X.  RR* ) --> ~P X )
26 xrex 10347 . . . 4  |-  RR*  e.  _V
27 xpexg 4799 . . . 4  |-  ( ( X  e.  dom  * Met  /\  RR*  e.  _V )  ->  ( X  X.  RR* )  e.  _V )
2817, 26, 27sylancl 643 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( X  X.  RR* )  e.  _V )
29 pwexg 4193 . . . 4  |-  ( X  e.  dom  * Met  ->  ~P X  e.  _V )
3017, 29syl 15 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ~P X  e.  _V )
31 fex2 5367 . . 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 )
3225, 28, 30, 31syl3anc 1182 . 2  |-  ( D  e.  ( * Met `  X )  ->  (
x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } )  e.  _V )
332, 13, 15, 32fvmptd 5568 1  |-  ( D  e.  ( * Met `  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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544   {crab 2548   _Vcvv 2789    C_ wss 3153   ~Pcpw 3626   U.cuni 3828   class class class wbr 4024    e. cmpt 4078    X. cxp 4686    dom cdm 4688   ran crn 4689   -->wf 5217   ` cfv 5221  (class class class)co 5820    e. cmpt2 5822   RR*cxr 8862    < clt 8863   * Metcxmt 16365   ballcbl 16367
This theorem is referenced by:  blval  17944  blf  17957
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-map 6770  df-xr 8867  df-xmet 16369  df-bl 16371
  Copyright terms: Public domain W3C validator