defpred S₁[ object ] means ( $1 is MSSubset of U0 & ( for B being MSSubset of U0 st B = $1 holds
B is opers_closed ) );
consider X being set such that
A1: for x being object holds
( x in X iff ( x in Funcs ( the carrier of S,(bool (Union the Sorts of U0))) & S₁[x] ) ) from XBOOLE_0:sch 1();
take X ; :: thesis: for x being object holds
( x in X iff ( x in Funcs ( the carrier of S,(bool (Union the Sorts of U0))) & x is MSSubset of U0 & ( for B being MSSubset of U0 st B = x holds
B is opers_closed ) ) )
thus for x being object holds
( x in X iff ( x in Funcs ( the carrier of S,(bool (Union the Sorts of U0))) & x is MSSubset of U0 & ( for B being MSSubset of U0 st B = x holds
B is opers_closed ) ) ) by A1; :: thesis: verum