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

Theorem dfec2 6867
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 6866 . 2  |-  [ A ] R  =  ( R " { A }
)
2 imasng 5185 . 2  |-  ( A  e.  V  ->  ( R " { A }
)  =  { y  |  A R y } )
31, 2syl5eq 2448 1  |-  ( A  e.  V  ->  [ A ] R  =  {
y  |  A R y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {cab 2390   {csn 3774   class class class wbr 4172   "cima 4840   [cec 6862
This theorem is referenced by:  eqglact  14946  tgpconcompeqg  18094  fvline  25982  ellines  25990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866
  Copyright terms: Public domain W3C validator