reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem Th9:
  y in Im(R,x) iff [x,y] in R
proof
  thus y in Im(R,x) implies [x,y] in R
  proof
    assume y in Im(R,x);
    then ex a being object st ( [a,y] in R)&( a in {x}) by RELAT_1:def 13;
    hence thesis by TARSKI:def 1;
  end;
  assume
A1: [x,y] in R;
  x in {x} by TARSKI:def 1;
  hence thesis by A1,RELAT_1:def 13;
end;
