reserve i,j,k,n for Nat;
reserve x,x1,x2,x3,y1,y2,y3 for set;

theorem Th8:
  for a,b being set holds (a .--> b)" = b .--> a
proof
  let a,b be set;
  set f = a .--> b, g = b .--> a;
A1: for y,x be object holds y in rng f & x = g.y iff x in dom f & y = f.x
  proof
    let y,x be object;
    thus y in rng f & x = g.y implies x in dom f & y = f.x
    proof
      assume that
A2:   y in rng f and
A3:   x = g.y;
A4:   y in {b} by A2,FUNCOP_1:8;
      then
A5:   x = g.b by A3,TARSKI:def 1
        .= a by FUNCOP_1:72;
      then
A6:   x in {a} by TARSKI:def 1;
      f.x = b by A5,FUNCOP_1:72
        .= y by A4,TARSKI:def 1;
      hence thesis by A6;
    end;
    assume that
A7: x in dom f and
A8: y = f.x;
A9: x in {a} by A7;
    then
A10: y = f.a by A8,TARSKI:def 1
      .= b by FUNCOP_1:72;
    then
A11: y in {b} by TARSKI:def 1;
    g.y = a by A10,FUNCOP_1:72
      .= x by A9,TARSKI:def 1;
    hence thesis by A11,FUNCOP_1:8;
  end;
  dom g = {b}
    .= rng f by FUNCOP_1:8;
  hence thesis by A1,FUNCT_1:32;
end;
