consider X being set such that
A1: for x being object holds
( x in X iff ( x in PFuncs (D,NAT) & S₁[x] ) ) from XBOOLE_0:sch 1();
take X ; :: thesis: for f being object holds
( f in X iff f is PartFunc of D,NAT )
let f be object ; :: thesis: ( f in X iff f is PartFunc of D,NAT )
thus ( f in X implies f is PartFunc of D,NAT ) by A1; :: thesis: ( f is PartFunc of D,NAT implies f in X )
assume A2: f is PartFunc of D,NAT ; :: thesis: f in X
then f in PFuncs (D,NAT) by PARTFUN1:45;
hence f in X by A1, A2; :: thesis: verum