defpred S₁[ object ] means ( $1 in X & the Target of G . $1 = v );
consider IT being set such that
A1: for x being object holds
( x in IT iff ( x in the carrier' of G & S₁[x] ) ) from XBOOLE_0:sch 1();
take IT ; :: thesis: ( IT is Subset of the carrier' of G & ( for e being set holds
( e in IT iff ( e in the carrier' of G & e in X & the Target of G . e = v ) ) ) )
for x being object st x in IT holds
x in the carrier' of G by A1;
hence IT is Subset of the carrier' of G by TARSKI:def 3; :: thesis: for e being set holds
( e in IT iff ( e in the carrier' of G & e in X & the Target of G . e = v ) )
thus for e being set holds
( e in IT iff ( e in the carrier' of G & e in X & the Target of G . e = v ) ) by A1; :: thesis: verum