reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem Th71:
  y in rng R iff R"{y} <> {}
proof
  thus y in rng R implies R"{y} <> {}
  proof
    assume y in rng R;
    then
A1: ex x being object st [x,y] in R by XTUPLE_0:def 13;
    y in {y} by TARSKI:def 1;
    hence thesis by A1,RELAT_1:def 14;
  end;
  assume R"{y} <> {};
  then consider x being object such that
A2: x in R"{y} by XBOOLE_0:def 1;
  consider z being object such that
A3: [x,z] in R and
A4: z in {y} by A2,RELAT_1:def 14;
  z = y by A4,TARSKI:def 1;
  hence thesis by A3,XTUPLE_0:def 13;
end;
