 reserve X,a,b,c,x,y,z,t for set;
 reserve R for Relation;

theorem Th21:
  for f being Function st dom f is subset-closed &
  dom f is d.union-closed & f is d.union-distributive holds
  for a, y being set st
  a in dom f & y in f.a ex b being set st b is finite & b c= a & y in f.b
proof
    let f be Function such that
A1: dom f is subset-closed;
    assume that
A2: dom f is d.union-closed and
AA: f is d.union-distributive;
    reconsider C = dom f as d.union-closed subset-closed set by A1,A2;
    let a, y be set;
    assume that
A3: a in dom f and
A4: y in f.a;
    reconsider A = {b where b is Subset of a: b is finite} as set;
A5: A is c=directed
    proof
      let Y be finite Subset of A;
      take union Y;
      now
        let x be set;
        assume x in Y;
        then x in A;
        then ex c being Subset of a st x = c & c is finite;
        hence x c= a;
      end;
      then
A6:   union Y c= a by ZFMISC_1:76;
      now
        let b be set;
        assume b in Y;
        then b in A;
        then ex c being Subset of a st b = c & c is finite;
        hence b is finite;
      end;
      then union Y is finite by FINSET_1:7;
      hence thesis by A6;
    end;
A7: union A = a
    proof
      thus union A c= a
      proof
        let x be object;
        assume x in union A;
        then consider b being set such that
A8:     x in b and
A9:     b in A by TARSKI:def 4;
        ex c being Subset of a st b = c & c is finite by A9;
        hence thesis by A8;
      end;
      let x be object;
      assume x in a;
      then {x} c= a by ZFMISC_1:31;
      then x in {x} & {x} in A by TARSKI:def 1;
      hence thesis by TARSKI:def 4;
    end;
A10: A c= C
    proof
      let x be object;
      assume x in A;
      then ex b being Subset of a st x = b & b is finite;
      hence thesis by A3,CLASSES1:def 1;
    end;
    f.union A = union (f.:A) by A3,A7,A5,A10,AA,COHSP_1:def 10;
    then consider B being set such that
A11: y in B and
A12: B in f.:A by A4,A7,TARSKI:def 4;
    consider b being object such that
    b in dom f and
A13: b in A and
A14: B = f.b by A12,FUNCT_1:def 6;
     reconsider bb = b as set by TARSKI:1;
    take bb;
    ex c being Subset of a st b = c & c is finite by A13;
    hence bb is finite & bb c= a & y in f.bb by A11,A14;
end;
