
theorem Th1:
  for X,Y be non empty set, f be Function of X,Y st f is bijective
    holds .:f is bijective & for s be Subset of X holds .:f.s = f.:s
proof
    let X,Y be non empty set, f be Function of X,Y;
    assume
A1: f is bijective;
A2: dom f = X by FUNCT_2:def 1;
    for y being object st y in bool Y
     ex x being object st x in bool X & y = .:f.x
    proof
     let y be object;
     assume y in bool Y; then
     reconsider Z = y as Subset of rng f by A1,FUNCT_2:def 3;
A3:  f.:(f"Z) = Z by FUNCT_1:77;
     take x = f"Z;
     thus thesis by A2,A3,FUNCT_3:def 1;
    end; then
    rng .:f = bool Y by FUNCT_2:10; then
    .:f is onto by FUNCT_2:def 3;
    hence .:f is bijective by A1;
    thus thesis by A2,FUNCT_3:def 1;
end;
