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 PartFunc of {x},Y holds f is one-to-one
proof
  let f be PartFunc of {x},Y;
  let x1,x2 be object;
  assume that
A1: x1 in dom f and
A2: x2 in dom f;
  dom f<>{} implies x1 = x & x2 = x by A1,A2,TARSKI:def 1;
  hence thesis by A1;
end;
