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;
reserve e,u for object,
  A for Subset of X;

theorem
  for f being Function st X c= dom f & f is one-to-one holds f"(f.:X) = X
proof
  let f be Function such that
A1: X c= dom f and
A2: f is one-to-one;
  thus f"(f.:X) c= X by A2,Th81;
  let x be object;
  assume
A3: x in X;
  then f.x in f.:X by A1,Def6;
  hence thesis by A1,A3,Def7;
end;
