reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve T for Nat;

theorem L:
  for X being non empty set
  for t being object
  for f being Function st dom f = X holds
  {w where w is Element of X: f.w = t} = Coim(f,t)
proof
  let X be non empty set;
  let t be object;
  let f be Function such that
AA: dom f = X;
  set A = {w where w is Element of X: f.w = t};
A0: t in {t} by TARSKI:def 1;
  thus A c= Coim(f,t)
  proof
    let x be object;
    assume x in A;
    then ex w being Element of X st x = w & f.w = t;
    then [x,t] in f by AA,FUNCT_1:1;
    hence thesis by A0,RELAT_1:def 14;
  end;
  let x be object;
  assume x in Coim(f,t);
  then consider y being object such that
A1: [x,y] in f and
A2: y in {t} by RELAT_1:def 14;
A4: y = t by A2,TARSKI:def 1;
A3: x in dom f by A1,XTUPLE_0:def 12;
    f.x = t by A1,A4,FUNCT_1:1;
  hence thesis by AA,A3;
end;
