reserve X,Y,Z,X1,X2,Y1,Y2 for set, x,y,z,t,x1,x2 for object,
  f,g,h,f1,f2,g1,g2 for Function;

theorem
 for x being object holds
  X,Funcs({x},X) are_equipotent & card X = card Funcs({x},X)
proof let x be object;
  deffunc F(object) = {x} --> $1;
  consider f such that
A1: dom f = X &
for y being object st y in X holds f.y = F(y) from FUNCT_1:sch 3;
A2: x in {x} by TARSKI:def 1;
  thus X,Funcs({x},X) are_equipotent
  proof
    take f;
    thus f is one-to-one
    proof
      let y,z be object;
      assume y in dom f & z in dom f;
      then
A3:   f.y = {x} --> y & f.z = {x} --> z by A1;
      ({x} --> y).x = y by A2,FUNCOP_1:7;
      hence thesis by A2,A3,FUNCOP_1:7;
    end;
    thus dom f = X by A1;
    thus rng f c= Funcs({x},X)
    proof
      let y be object;
      assume y in rng f;
      then consider z being object such that
A4:   z in dom f & y = f.z by FUNCT_1:def 3;
A5:   dom({x} --> z) = {x} & rng({x} --> z) = {z} by FUNCOP_1:8,13;
      y = {x} --> z & {z} c= X by A1,A4,ZFMISC_1:31;
      hence thesis by A5,FUNCT_2:def 2;
    end;
    let y be object;
    assume y in Funcs({x},X);
    then consider g such that
A6: y = g and
A7: dom g = {x} and
A8: rng g c= X by FUNCT_2:def 2;
A9: g.x in {g.x} by TARSKI:def 1;
A10: rng g = {g.x} by A7,FUNCT_1:4;
    then g = {x} --> (g.x) by A7,FUNCOP_1:9;
    then f.(g.x) = g by A1,A8,A10,A9;
    hence thesis by A1,A6,A8,A10,A9,FUNCT_1:def 3;
  end;
  hence thesis by CARD_1:5;
end;
