reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem
  Union disjoin (Y --> X) = [:X,Y:]
proof
  set f = Y --> X;
A1: dom f = Y;
  thus Union disjoin f c= [:X,Y:]
  proof
    let x be object;
    assume x in Union disjoin f;
    then consider Z such that
A2: x in Z and
A3: Z in rng disjoin f by TARSKI:def 4;
    consider y being object such that
A4: y in dom disjoin f and
A5: Z = (disjoin f).y by A3,FUNCT_1:def 3;
A6: y in Y by A1,A4,Def3;
    then
A7: Z = [:f.y,{y}:] by A1,A5,Def3;
A8: f.y = X by A6,FUNCOP_1:7;
    {y} c= Y by A6,ZFMISC_1:31;
    then Z c= [:X,Y:] by A7,A8,ZFMISC_1:95;
    hence thesis by A2;
  end;
  let x1,x2 be object;
  assume
A9: [x1,x2] in [:X,Y:];
  then
A10: x1 in X by ZFMISC_1:87;
A11: x2 in Y by A9,ZFMISC_1:87;
  then
A12: f.x2 = X by FUNCOP_1:7;
A13: (disjoin f).(x2) = [:f.x2,{x2}:] by A1,A11,Def3;
A14: x2 in dom disjoin f by A1,A11,Def3;
  x2 in {x2} by TARSKI:def 1;
  then
A15: [x1,x2] in [:f.x2,{x2}:] by A10,A12,ZFMISC_1:87;
  [:f.x2,{x2}:] in rng disjoin f by A13,A14,FUNCT_1:def 3;
  hence thesis by A15,TARSKI:def 4;
end;
