let A, B be set ; :: thesis: ( ( for x being set holds
( x in A iff f . x <> 0 ) ) & ( for x being set holds
( x in B iff f . x <> 0 ) ) implies A = B )
assume that
A3: for x being set holds
( x in A iff f . x <> 0 ) and
A4: for x being set holds
( x in B iff f . x <> 0 ) ; :: thesis: A = B
for x being set holds
( x in A iff x in B )

proof

let x be set ; :: thesis: ( x in A iff x in B )
( x in A iff f . x <> 0 ) by A3;
hence ( x in A iff x in B ) by A4; :: thesis: verum

end;

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