Could someone please check my work?

Suppose f: A B. Let D be a subset of B. Prove f-'(BJD) = Alf"' (D ). Show that f"' (BJD ) SANf-' (D ). Ler xE f-( BID). By The definition of preimage of BID in A f (x ) E BID Then "By definition of complement, f( x) EB and f(x) #D By the definition of preimage of Din A, > xef"( B) and XEF."(D) and XEA The? By The definition of complement, XEA( F- (D) Then by The definition of subset, Thus f" ( BID ) = Alf' CD ) . -> Show That Alf-'(D) = f-(BJD ). Ler x E Alf"(D). Then by The definition of complement 7 XEA and XE f"(D). Then by The definition of preimage of D and The definition of the In A image of A inB > f( x) EB and f( x ) $D Then by The definition of The preimage of Bing > xef" ( B) Then by The defining of Complement, = xE f" ( BID). Then by The definition of Subset, Thus , Alf -" (D ) = f " ( BID ) . -2 equivalent There fore, by and 2, and The definition of sets, f - ( B ID ) = A / f -" (D )