defpred S₁[ set ] means ex Z being set st
( $1 = Z & ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y ) );
consider XX being set such that
A8: for x being set holds
( x in XX iff ( x in bool ((union SFX) /\ (union SFY)) & S₁[x] ) ) from XFAMILY:sch 1();
take XX ; :: thesis: for Z being set holds
( Z in XX iff ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y ) )
for Z being set holds
( Z in XX iff ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y ) )

proof

let Z be set ; :: thesis: ( Z in XX iff ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y ) )

A9: now :: thesis: ( ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y ) implies Z in XX )

given X, Y being set such that A10: ( X in SFX & Y in SFY ) and
A11: Z = X /\ Y ; :: thesis: Z in XX
( X c= union SFX & Y c= union SFY ) by A10, ZFMISC_1:74;
then Z c= (union SFX) /\ (union SFY) by A11, XBOOLE_1:27;
hence Z in XX by A8, A10, A11; :: thesis: verum

end;

now :: thesis: ( Z in XX implies ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y ) )

assume Z in XX ; :: thesis: ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y )
then ex Z1 being set st
( Z = Z1 & ex X, Y being set st
( X in SFX & Y in SFY & Z1 = X /\ Y ) ) by A8;
hence ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y ) ; :: thesis: verum

end;

hence ( Z in XX iff ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y ) ) by A9; :: thesis: verum

end;

hence for Z being set holds
( Z in XX iff ex X, Y being set st
( X in SFX & Y in SFY & Z = X /\ Y ) ) ; :: thesis: verum