reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve S for non empty Subset of REAL;
reserve r for Real;
reserve T for Nat;

theorem
  for A, I, y being non empty set
  for F being Function of A,I holds
   { z where z is Element of A : F.z in y } = F"y
   proof
    let A, I, y be non empty set;
    let F be Function of A,I;
    for x being object holds
    x in { z where z is Element of A : F.z in y }
     iff x in F"y
    proof
    let x be object;
    hereby
     assume x in { z where z is Element of A : F.z in y };
     then consider z being Element of A such that
     A1: x = z & F.z in y;
     z in A;
     then z in dom F by FUNCT_2:def 1;
     hence x in F"y by FUNCT_1:def 7,A1;
    end;
    assume x in F"y;
    then x in dom F & F.x in y by FUNCT_1:def 7;
    hence x in { z where z is Element of A : F.z in y };
    end;
    hence thesis by TARSKI:2;
   end;
