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;

theorem
  f is one-to-one & dom f = rng g & rng f = dom g & (for x,y st x in dom
  f & y in dom g holds f.x = y iff g.y = x) implies g = f"
proof
  assume that
A1: f is one-to-one and
A2: dom f = rng g and
A3: rng f = dom g and
A4: for x,y st x in dom f & y in dom g holds f.x = y iff g.y = x;
A5: y in dom g implies g.y = (f").y
  proof
    assume
A6: y in dom g;
    then
A7: g.y in dom f by A2,Def3;
    then f.(g.y) = y by A4,A6;
    hence thesis by A1,A7,Th31;
  end;
  rng f = dom(f") by A1,Th31;
  hence thesis by A3,A5,Th2;
end;
