MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfec2 Unicode version

Theorem dfec2 6659
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 6658 . 2  |-  [ A ] R  =  ( R " { A }
)
2 imasng 5034 . 2  |-  ( A  e.  V  ->  ( R " { A }
)  =  { y  |  A R y } )
31, 2syl5eq 2328 1  |-  ( A  e.  V  ->  [ A ] R  =  {
y  |  A R y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1685   {cab 2270   {csn 3641   class class class wbr 4024   "cima 4691   [cec 6654
This theorem is referenced by:  eqglact  14664  tgpconcompeqg  17790  fvline  24177  ellines  24185
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 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-ec 6658
  Copyright terms: Public domain W3C validator