let A, X1, Y1, B, X2, Y2 be set ; :: thesis: ( A c= [:X1,Y1:] & B c= [:X2,Y2:] & ( for x, y being set holds
( [x,y] in A iff [x,y] in B ) ) implies A = B )
assume that
A1: ( A c= [:X1,Y1:] & B c= [:X2,Y2:] ) and
A2: for x, y being set holds
( [x,y] in A iff [x,y] in B ) ; :: thesis: A = B

now

let z be set ; :: thesis: ( z in A iff z in B )
( ( z in A implies ex x, y being set st
( x in X1 & y in Y1 & z = [x,y] ) ) & ( z in B implies ex x, y being set st
( x in X2 & y in Y2 & z = [x,y] ) ) ) by A1, Th103;
hence ( z in A iff z in B ) by A2; :: thesis: verum

end;

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