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Theorem blfvalps 19963
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 17817 . . 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 5045 . . . . 5  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5047 . . . 4  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 psmetdmdm 19886 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
65eqcomd 2448 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  dom  dom  D  =  X )
74, 6sylan9eqr 2497 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
8 eqidd 2444 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  RR*  =  RR* )
9 simpr 461 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  d  =  D )
109oveqd 6113 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x d y )  =  ( x D y ) )
1110breq1d 4307 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
( x d y )  <  r  <->  ( x D y )  < 
r ) )
127, 11rabeqbidv 2972 . . 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 6153 . 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 2986 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  _V )
15 ssrab2 3442 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
16 elfvdm 5721 . . . . . . . 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 4460 . . . . . . 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 2813 . . . 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 2443 . . . . 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 6646 . . . 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 10993 . . . 4  |-  RR*  e.  _V
26 xpexg 6512 . . . 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 4481 . . . 4  |-  ( X  e.  dom PsMet  ->  ~P X  e.  _V )
2916, 28syl 16 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ~P X  e.  _V )
30 fex2 6537 . . 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 1218 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } )  e. 
_V )
322, 13, 14, 31fvmptd 5784 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 1369    e. wcel 1756   A.wral 2720   {crab 2724   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   class class class wbr 4297    e. cmpt 4355    X. cxp 4843   dom cdm 4845   -->wf 5419   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   RR*cxr 9422    < clt 9423  PsMetcpsmet 17805   ballcbl 17808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-map 7221  df-xr 9427  df-psmet 17814  df-bl 17817
This theorem is referenced by:  blfval  19964  blvalps  19965  blfps  19986
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