reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;

theorem
  for f being Function st dom f = {x} & x in X & f.x in Y holds f is
  PartFunc of X,Y
proof
  let f be Function;
  assume that
A1: dom f = {x} and
A2: x in X and
A3: f.x in Y;
  rng f = {f.x} by A1,FUNCT_1:4;
  then
A4: rng f c= Y by A3,ZFMISC_1:31;
  dom f c= X by A1,A2,ZFMISC_1:31;
  hence thesis by A4,RELSET_1:4;
end;
