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 Th58:
  x in dom f implies Im(f,x) = {f.x}
proof
  assume
A1: x in dom f;
  for y being object holds y in f.:{x} iff y in {f.x}
  proof let y be object;
    thus y in f.:{x} implies y in {f.x}
    proof
      assume y in f.:{x};
      then consider z being object such that
      z in dom f and
A2:   z in {x} and
A3:   y = f.z by Def6;
      z = x by 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 {x} by TARSKI:def 1;
    hence thesis by A1,A4,Def6;
  end;
  hence thesis by TARSKI:2;
end;
