let A, B be set ; :: thesis: ( ( for x being set holds
( x in A iff ( x in dom f & f . x <> 0. S ) ) ) & ( for x being set holds
( x in B iff ( x in dom f & f . x <> 0. S ) ) ) implies A = B )
assume that
A2: for x being set holds
( x in A iff ( x in dom f & f . x <> 0. S ) ) and
A3: for x being set holds
( x in B iff ( x in dom f & f . x <> 0. S ) ) ; :: thesis: A = B

now :: thesis: for x being object holds
( x in A iff x in B )

let x be object ; :: thesis: ( x in A iff x in B )
( x in A iff ( x in dom f & f . x <> 0. S ) ) by A2;
hence ( x in A iff x in B ) by A3; :: thesis: verum

end;

hence A = B by TARSKI:2; :: thesis: verum