ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfec2 Unicode version

Theorem dfec2 6400
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 6399 . 2  |-  [ A ] R  =  ( R " { A }
)
2 imasng 4874 . 2  |-  ( A  e.  V  ->  ( R " { A }
)  =  { y  |  A R y } )
31, 2syl5eq 2162 1  |-  ( A  e.  V  ->  [ A ] R  =  {
y  |  A R y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   {cab 2103   {csn 3497   class class class wbr 3899   "cima 4512   [cec 6395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-ec 6399
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator