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 st f is one-to-one holds f" is PartFunc of Y,X
proof
  let f be PartFunc of X,Y such that
A1: f is one-to-one;
  rng f c= Y by RELAT_1:def 19;
  then
A2: dom(f") c= Y by A1,FUNCT_1:33;
  dom f c= X;
  then rng(f") c= X by A1,FUNCT_1:33;
  hence thesis by A2,RELSET_1:4;
end;
