
theorem Th4:
  for A being set, F being Subset-Family of A, f be one-to-one Function holds
    f.: meet F = meet {f.:X where X is Subset of A : X in F}
proof
  let A be set, F be Subset-Family of A, f be one-to-one Function;
  set S = {f.:X where X is Subset of A : X in F};
  A7: meet S c= f.: meet F
  proof
    let y be object;
    assume A1: y in meet S;
    then consider z being object such that
      A2: z in S by XBOOLE_0:def 1, SETFAM_1:1;
    consider X0 being Subset of A such that
      A3: z = f.:X0 & X0 in F by A2;
    A4: y in f.:X0 by A1, A2, A3, SETFAM_1:def 1;
    ex x being object st x in dom f & x in meet F & y = f.x
    proof
      consider x0 being object such that
        A5: x0 in dom f & x0 in X0 & y = f.x0 by A4, FUNCT_1:def 6;
      take x0;
      for X being set st X in F holds x0 in X
      proof
        let X be set;
        assume X in F;
        then f.:X in S;
        then y in f.:X by A1, SETFAM_1:def 1;
        then consider x being object such that
          A6: x in dom f & x in X & y = f.x by FUNCT_1:def 6;
        thus thesis by A5, A6, FUNCT_1:def 4;
      end;
      hence thesis by A3, A5, SETFAM_1:def 1;
    end;
    hence thesis by FUNCT_1:def 6;
  end;
  f.: meet F c= meet S by Th3;
  hence thesis by A7, XBOOLE_0:def 10;
end;
