defpred S₁[ set ] means ex y being set st
( [$1,y] in R & y in Y );
consider X being set such that
A1: for x being set holds
( x in X iff ( x in dom R & S₁[x] ) ) from XBOOLE_0:sch 1();
take X ; :: thesis: for x being set holds
( x in X iff ex y being set st
( [x,y] in R & y in Y ) )
let x be set ; :: thesis: ( x in X iff ex y being set st
( [x,y] in R & y in Y ) )
thus ( x in X implies ex y being set st
( [x,y] in R & y in Y ) ) by A1; :: thesis: ( ex y being set st
( [x,y] in R & y in Y ) implies x in X )
given y being set such that A2: [x,y] in R and
A3: y in Y ; :: thesis: x in X
x in dom R by A2, Def4;
hence x in X by A1, A2, A3; :: thesis: verum