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 Th22:
  x in Union disjoin f iff x`2 in dom f & x`1 in f.(x`2) & x = [x`1,x`2]
proof
  thus x in Union disjoin f implies
  x`2 in dom f & x`1 in f.(x`2) & x = [x`1,x`2]
  proof
    assume x in Union disjoin f;
    then consider X such that
A1: x in X and
A2: X in rng disjoin f by TARSKI:def 4;
    consider y being object such that
A3: y in dom disjoin f and
A4: X = (disjoin f).y by A2,FUNCT_1:def 3;
A5: y in dom f by A3,Def3;
    then
A6: X = [:f.y,{y}:] by A4,Def3;
    then
A7: x`1 in f.y by A1,MCART_1:10;
    x`2 in {y} by A1,A6,MCART_1:10;
    hence thesis by A1,A5,A6,A7,MCART_1:21,TARSKI:def 1;
  end;
  assume that
A8: x`2 in dom f and
A9: x`1 in f.(x`2) and
A10: x = [x`1,x`2];
A11: (disjoin f).(x`2) = [:f.(x`2),{x`2}:] by A8,Def3;
A12: dom f = dom disjoin f by Def3;
  x`2 in {x`2} by TARSKI:def 1;
  then
A13: x in [:f.(x`2),{x`2}:] by A9,A10,ZFMISC_1:87;
  [:f.(x`2),{x`2}:] in rng disjoin f by A8,A11,A12,FUNCT_1:def 3;
  hence thesis by A13,TARSKI:def 4;
end;
