reserve G for _Graph;
reserve V for non empty set, E for Relation of V;

theorem Th75:
  for Y being set holds Y|`E = createGraph(V,E).edgesInto(Y)
proof
  let Y be set;
  set G = createGraph(V,E);
  now
    let z be object;
    hereby
      assume A1: z in Y|`E;
      then consider x,y being object such that
        A2: z = [x,y] by RELAT_1:def 1;
      A3: y in Y & [x,y] in E by A1, A2, RELAT_1:def 12;
      then [x,y] DJoins x,y,G by Th63;
      then [x,y] in the_Edges_of G & (the_Target_of G).[x,y] = y
        by GLIB_000:def 14;
      hence z in G.edgesInto(Y) by A2, A3, GLIB_000:def 26;
    end;
    set x = (the_Source_of G).z, y = (the_Target_of G).z;
    assume z in G.edgesInto(Y);
    then A4: z in the_Edges_of G & y in Y by GLIB_000:def 26;
    then z DJoins x,y,G by GLIB_000:def 14;
    then z = [x,y] by Th64;
    hence z in Y|`E by A4, RELAT_1:def 12;
  end;
  hence thesis by TARSKI:2;
end;
