reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th159:
  [x,y] in R iff y in Im(R,x)
  proof
    thus [x,y] in R implies y in Im(R,x)
    proof
      x in {x} by TARSKI:def 1;
      hence thesis by Def11;
    end;
    assume y in Im(R,x);
    then ex z st [z,y] in R & z in {x} by Def11;
    hence [x,y] in R by TARSKI:def 1;
  end;
