
theorem
  for A, B, a, b being set, f being Function of A,B st a in A & b in B
  holds f +* (a .--> b) is Function of A,B
proof
  let A, B, a, b be set, f be Function of A,B such that
A1: a in A and
A2: b in B;
A3: {a} c= A by A1,ZFMISC_1:31;
  set g = a .--> b;
  set f1 = f +* g;
  rng g = {b} by FUNCOP_1:8;
  then rng g c= B by A2,ZFMISC_1:31;
  then
A4: rng f1 c= rng f \/ rng g & rng f \/ rng g c= B \/ B by FUNCT_4:17
,XBOOLE_1:13;
  dom f1 = dom f \/ dom g by FUNCT_4:def 1
    .= A \/ dom g by A2,FUNCT_2:def 1
    .= A \/ {a}
    .= A by A3,XBOOLE_1:12;
  hence thesis by A4,FUNCT_2:2,XBOOLE_1:1;
end;
