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Theorem dfec2 6663
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 6662 . 2  |-  [ A ] R  =  ( R " { A }
)
2 imasng 5035 . 2  |-  ( A  e.  V  ->  ( R " { A }
)  =  { y  |  A R y } )
31, 2syl5eq 2327 1  |-  ( A  e.  V  ->  [ A ] R  =  {
y  |  A R y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   {csn 3640   class class class wbr 4023   "cima 4692   [cec 6658
This theorem is referenced by:  eqglact  14668  tgpconcompeqg  17794  fvline  24767  ellines  24775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662
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