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Theorem elblps 21016
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 21014 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P ( ball `  D
) R )  =  { x  e.  X  |  ( P D x )  <  R } )
21eleq2d 2527 . 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 6304 . . . 4  |-  ( x  =  A  ->  ( P D x )  =  ( P D A ) )
43breq1d 4466 . . 3  |-  ( x  =  A  ->  (
( P D x )  <  R  <->  ( P D A )  <  R
) )
54elrab 3257 . 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 973    = wceq 1395    e. wcel 1819   {crab 2811   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RR*cxr 9644    < clt 9645  PsMetcpsmet 18529   ballcbl 18532
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 18538  df-bl 18541
This theorem is referenced by:  elbl2ps  21018  xblpnfps  21024  xblss2ps  21030  xblcntrps  21039  blssps  21053
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