let X1, X2 be set ; :: thesis: ( ( for x being object holds
( x in X1 iff ex y being object st
( [x,y] in R & y in Y ) ) ) & ( for x being object holds
( x in X2 iff ex y being object st
( [x,y] in R & y in Y ) ) ) implies X1 = X2 )
assume that
A4: for x being object holds
( x in X1 iff ex y being object st
( [x,y] in R & y in Y ) ) and
A5: for x being object holds
( x in X2 iff ex y being object st
( [x,y] in R & y in Y ) ) ; :: thesis: X1 = X2

now :: thesis: for x being object holds
( x in X1 iff x in X2 )

let x be object ; :: thesis: ( x in X1 iff x in X2 )
( x in X1 iff ex y being object st
( [x,y] in R & y in Y ) ) by A4;
hence ( x in X1 iff x in X2 ) by A5; :: thesis: verum

end;

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