reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;

theorem
  for S,X being set, f being Function of S,X, A being Subset of X st X =
  {} implies S = {} holds (f"A)` = f"(A`)
proof
  let S,X be set, f be Function of S,X, A be Subset of X such that
A1: X = {} implies S = {};
  A /\ A` = {} by XBOOLE_0:def 7,XBOOLE_1:79;
  then f"A /\ f"(A`) = f"({}X) by FUNCT_1:68
    .= {};
  then
A2: f"A misses f"(A`);
  f"A \/ f"(A`) = f"(A \/ A`) by RELAT_1:140
    .= f"[#]X by SUBSET_1:10
    .= [#]S by A1,Th39;
  then (f"A)` /\ (f"(A`))` = ([#]S)` by XBOOLE_1:53
    .= {}S by XBOOLE_1:37;
  then (f"A)` misses (f"(A`))`;
  hence thesis by A2,SUBSET_1:25;
end;
