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 Th4:
  dom f = {x} implies rng f = {f.x}
proof
  assume
A1: dom f = {x};
  for y being object holds y in rng f iff y in {f.x}
  proof let y be object;
    thus y in rng f implies y in {f.x}
    proof
      assume y in rng f;
      then consider z being object such that
A2:   z in dom f and
A3:   y = f.z by Def3;
      z = x by A1,A2,TARSKI:def 1;
      hence thesis by A3,TARSKI:def 1;
    end;
    assume y in {f.x};
    then
A4: y = f.x by TARSKI:def 1;
    x in dom f by A1,TARSKI:def 1;
    hence thesis by A4,Def3;
  end;
  hence thesis by TARSKI:2;
end;
