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Theorem elblps 20618
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 20616 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P ( ball `  D
) R )  =  { x  e.  X  |  ( P D x )  <  R } )
21eleq2d 2530 . 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 6283 . . . 4  |-  ( x  =  A  ->  ( P D x )  =  ( P D A ) )
43breq1d 4450 . . 3  |-  ( x  =  A  ->  (
( P D x )  <  R  <->  ( P D A )  <  R
) )
54elrab 3254 . 2  |-  ( A  e.  { x  e.  X  |  ( P D x )  < 
R }  <->  ( A  e.  X  /\  ( P D A )  < 
R ) )
62, 5syl6bb 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {crab 2811   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   RR*cxr 9616    < clt 9617  PsMetcpsmet 18166   ballcbl 18169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-xr 9621  df-psmet 18175  df-bl 18178
This theorem is referenced by:  elbl2ps  20620  xblpnfps  20626  xblss2ps  20632  xblcntrps  20641  blssps  20655
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