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Theorem elblps 13821
Description: Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
elblps  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( A  e.  ( P
( ball `  D ) R )  <->  ( A  e.  X  /\  ( P D A )  < 
R ) ) )

Proof of Theorem elblps
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 blvalps 13819 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P ( ball `  D
) R )  =  { x  e.  X  |  ( P D x )  <  R } )
21eleq2d 2247 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( A  e.  ( P
( ball `  D ) R )  <->  A  e.  { x  e.  X  | 
( P D x )  <  R }
) )
3 oveq2 5882 . . . 4  |-  ( x  =  A  ->  ( P D x )  =  ( P D A ) )
43breq1d 4013 . . 3  |-  ( x  =  A  ->  (
( P D x )  <  R  <->  ( P D A )  <  R
) )
54elrab 2893 . 2  |-  ( A  e.  { x  e.  X  |  ( P D x )  < 
R }  <->  ( A  e.  X  /\  ( P D A )  < 
R ) )
62, 5bitrdi 196 1  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( A  e.  ( P
( ball `  D ) R )  <->  ( A  e.  X  /\  ( P D A )  < 
R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   {crab 2459   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   RR*cxr 7989    < clt 7990  PsMetcpsmet 13370   ballcbl 13373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-map 6649  df-pnf 7992  df-mnf 7993  df-xr 7994  df-psmet 13378  df-bl 13381
This theorem is referenced by:  elbl2ps  13823  xblpnfps  13829  xblss2ps  13835  xblcntrps  13844  blssps  13858
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