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Theorem dfec2 6750
Description: Alternate definition of  R-coset of  A. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
dfec2  |-  ( A  e.  V  ->  [ A ] R  =  {
y  |  A R y } )
Distinct variable groups:    y, A    y, R
Allowed substitution hint:    V( y)

Proof of Theorem dfec2
StepHypRef Expression
1 df-ec 6749 . 2  |-  [ A ] R  =  ( R " { A }
)
2 imasng 5117 . 2  |-  ( A  e.  V  ->  ( R " { A }
)  =  { y  |  A R y } )
31, 2syl5eq 2402 1  |-  ( A  e.  V  ->  [ A ] R  =  {
y  |  A R y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   {cab 2344   {csn 3716   class class class wbr 4104   "cima 4774   [cec 6745
This theorem is referenced by:  eqglact  14767  tgpconcompeqg  17896  fvline  25326  ellines  25334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-ec 6749
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