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Theorem blfval 17947
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 16375 . . 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 4879 . . . . 5  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 4881 . . . 4  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 xmetdmdm 17900 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  X  =  dom  dom  D )
65eqcomd 2288 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  X )
74, 6sylan9eqr 2337 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
8 eqidd 2284 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  RR*  =  RR* )
9 simpr 447 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  d  =  D )
109oveqd 5875 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( x
d y )  =  ( x D y ) )
1110breq1d 4033 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( (
x d y )  <  r  <->  ( x D y )  < 
r ) )
127, 11rabeqbidv 2783 . . 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 5910 . 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 5551 . . 3  |-  ( * Met `  X ) 
C_  U. ran  * Met
1514sseli 3176 . 2  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
16 ssrab2 3258 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
17 elfvdm 5554 . . . . . . . 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 4174 . . . . . . 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 2635 . . . 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 2283 . . . . 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 6191 . . . 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 10351 . . . 4  |-  RR*  e.  _V
27 xpexg 4800 . . . 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 4194 . . . 4  |-  ( X  e.  dom  * Met  ->  ~P X  e.  _V )
3017, 29syl 15 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ~P X  e.  _V )
31 fex2 5401 . . 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 5606 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 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   RR*cxr 8866    < clt 8867   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  blval  17948  blf  17961
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-xr 8871  df-xmet 16373  df-bl 16375
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