ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  blfvalps Unicode version

Theorem blfvalps 12313
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 11941 . . 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 4677 . . . . 5  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 4679 . . . 4  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 psmetdmdm 12252 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
65eqcomd 2105 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  dom  dom  D  =  X )
74, 6sylan9eqr 2154 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
8 eqidd 2101 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  RR*  =  RR* )
9 simpr 109 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  d  =  D )
109oveqd 5723 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
x d y )  =  ( x D y ) )
1110breq1d 3885 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
( x d y )  <  r  <->  ( x D y )  < 
r ) )
127, 11rabeqbidv 2636 . . 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 5765 . 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 2652 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  _V )
15 ssrab2 3129 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
16 psmetrel 12250 . . . . . . . . 9  |-  Rel PsMet
17 relelfvdm 5385 . . . . . . . . 9  |-  ( ( Rel PsMet  /\  D  e.  (PsMet `  X ) )  ->  X  e.  dom PsMet )
1816, 17mpan 418 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
1918adantr 272 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
x  e.  X  /\  r  e.  RR* ) )  ->  X  e.  dom PsMet )
20 elpw2g 4021 . . . . . . 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 2473 . . . 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 2100 . . . . 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 6029 . . . 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 9480 . . . 4  |-  RR*  e.  _V
28 xpexg 4591 . . . 4  |-  ( ( X  e.  dom PsMet  /\  RR*  e.  _V )  ->  ( X  X.  RR* )  e.  _V )
2918, 27, 28sylancl 407 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( X  X.  RR* )  e.  _V )
3018pwexd 4045 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ~P X  e.  _V )
31 fex2 5227 . . 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 1184 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } )  e. 
_V )
332, 13, 14, 32fvmptd 5434 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 1299    e. wcel 1448   A.wral 2375   {crab 2379   _Vcvv 2641    C_ wss 3021   ~Pcpw 3457   class class class wbr 3875    |-> cmpt 3929    X. cxp 4475   dom cdm 4477   Rel wrel 4482   -->wf 5055   ` cfv 5059  (class class class)co 5706    e. cmpo 5708   RR*cxr 7671    < clt 7672  PsMetcpsmet 11930   ballcbl 11933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-psmet 11938  df-bl 11941
This theorem is referenced by:  blfval  12314  blvalps  12316  blfps  12337
  Copyright terms: Public domain W3C validator