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Theorem blfvalps 21011
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 18540 . . 3  |-  ball  =  ( d  e.  _V  |->  ( x  e.  dom  dom  d ,  r  e. 
RR*  |->  { y  e. 
dom  dom  d  |  ( x d y )  <  r } ) )
21a1i 11 . 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 5213 . . . . 5  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5215 . . . 4  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 psmetdmdm 20934 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
65eqcomd 2465 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  dom  dom  D  =  X )
74, 6sylan9eqr 2520 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
8 eqidd 2458 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  RR*  =  RR* )
9 simpr 461 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  d  =  D )
109oveqd 6313 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x d y )  =  ( x D y ) )
1110breq1d 4466 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
( x d y )  <  r  <->  ( x D y )  < 
r ) )
127, 11rabeqbidv 3104 . . 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, 12mpt2eq123dv 6358 . 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 3118 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  _V )
15 ssrab2 3581 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
16 elfvdm 5898 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
1716adantr 465 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  r  e.  RR* ) )  ->  X  e.  dom PsMet )
18 elpw2g 4619 . . . . . . 7  |-  ( X  e.  dom PsMet  ->  ( { y  e.  X  | 
( x D y )  <  r }  e.  ~P X  <->  { y  e.  X  |  (
x D y )  <  r }  C_  X ) )
1917, 18syl 16 . . . . . 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 ) )
2015, 19mpbiri 233 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  r  e.  RR* ) )  ->  { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X )
2120ralrimivva 2878 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  X  A. r  e.  RR*  { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X )
22 eqid 2457 . . . . 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 } )
2322fmpt2 6866 . . . 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 )
2421, 23sylib 196 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } ) : ( X  X.  RR* )
--> ~P X )
25 xrex 11242 . . . 4  |-  RR*  e.  _V
26 xpexg 6601 . . . 4  |-  ( ( X  e.  dom PsMet  /\  RR*  e.  _V )  ->  ( X  X.  RR* )  e.  _V )
2716, 25, 26sylancl 662 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( X  X.  RR* )  e.  _V )
28 pwexg 4640 . . . 4  |-  ( X  e.  dom PsMet  ->  ~P X  e.  _V )
2916, 28syl 16 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ~P X  e.  _V )
30 fex2 6754 . . 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 )
3124, 27, 29, 30syl3anc 1228 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } )  e. 
_V )
322, 13, 14, 31fvmptd 5961 1  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RR*cxr 9644    < clt 9645  PsMetcpsmet 18528   ballcbl 18531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-xr 9649  df-psmet 18537  df-bl 18540
This theorem is referenced by:  blfval  21012  blvalps  21013  blfps  21034
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