defpred S₁[ object ] means ex f being Function st
( $1 = f & dom f = X & rng f c= Y );
consider F being set such that
A1: for z being object holds
( z in F iff ( z in bool [:X,Y:] & S₁[z] ) ) from XBOOLE_0:sch 1();
take F ; :: thesis: for x being object holds
( x in F iff ex f being Function st
( x = f & dom f = X & rng f c= Y ) )
let z be object ; :: thesis: ( z in F iff ex f being Function st
( z = f & dom f = X & rng f c= Y ) )
thus ( z in F implies ex f being Function st
( z = f & dom f = X & rng f c= Y ) ) by A1; :: thesis: ( ex f being Function st
( z = f & dom f = X & rng f c= Y ) implies z in F )
given f being Function such that A2: z = f and
A3: ( dom f = X & rng f c= Y ) ; :: thesis: z in F
f c= [:X,Y:]

proof

let p be object ; :: according to TARSKI:def 3 :: thesis: ( not p in f or p in [:X,Y:] )
assume A4: p in f ; :: thesis: p in [:X,Y:]
then consider x, y being object such that
A5: p = [x,y] by RELAT_1:def 1;
reconsider y = y as set by TARSKI:1;
A6: x in dom f by A4, A5, XTUPLE_0:def 12;
then y = f . x by A4, A5, FUNCT_1:def 2;
then y in rng f by A6, FUNCT_1:def 3;
hence p in [:X,Y:] by A3, A5, A6, ZFMISC_1:def 2; :: thesis: verum

end;

hence z in F by A1, A2, A3; :: thesis: verum