let X1, X2 be set ; :: thesis: ( ( for x being set holds
( x in X1 iff ex y being set st [x,y] in R ) ) & ( for x being set holds
( x in X2 iff ex y being set st [x,y] in R ) ) implies X1 = X2 )
assume A2: ( ( for x being set holds
( x in X1 iff ex y being set st [x,y] in R ) ) & ( for x being set holds
( x in X2 iff ex y being set st [x,y] in R ) ) ) ; :: thesis: X1 = X2

now

let x be set ; :: thesis: ( x in X1 iff x in X2 )
( ( x in X1 implies ex y being set st [x,y] in R ) & ( ex y being set st [x,y] in R implies x in X1 ) & ( x in X2 implies ex y being set st [x,y] in R ) & ( ex y being set st [x,y] in R implies x in X2 ) ) by A2;
hence ( x in X1 iff x in X2 ) ; :: thesis: verum

end;

hence X1 = X2 by TARSKI:2; :: thesis: verum