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 T,S being non empty set, f being Function of T,S, P being
  Subset-Family of T holds union P c= union(f"(f.:P))
proof
  let T,S be non empty set;
  let f be Function of T,S;
  let P be Subset-Family of T;
  let x be object;
  assume x in union P;
  then consider A being set such that
A1: x in A and
A2: A in P by TARSKI:def 4;
A3: A c= T by A2;
  reconsider A as Subset of T by A2;
  reconsider A1 = f.:A as Subset of S;
  reconsider A2 = f"A1 as Subset of T;
  A c= dom f by A3,Def1;
  then
A4: A c= A2 by FUNCT_1:76;
  A1 in f.:P by A2,Def10;
  then A2 in f"(f.:P) by Def9;
  hence thesis by A1,A4,TARSKI:def 4;
end;
