defpred S₁[ object , object ] means for x1, y1 being object st $1 = [x1,y1] holds
P₁[x1,y1,$2];
A2: for x being object st x in [:F₁(),F₂():] holds
ex z being object st
( z in F₃() & S₁[x,z] )

proof

let x be object ; :: thesis: ( x in [:F₁(),F₂():] implies ex z being object st
( z in F₃() & S₁[x,z] ) )
assume x in [:F₁(),F₂():] ; :: thesis: ex z being object st
( z in F₃() & S₁[x,z] )
then consider x1, y1 being object such that
A3: ( x1 in F₁() & y1 in F₂() ) and
A4: x = [x1,y1] by ZFMISC_1:def 2;
consider z being object such that
A5: z in F₃() and
A6: P₁[x1,y1,z] by A1, A3;
take z ; :: thesis: ( z in F₃() & S₁[x,z] )
thus z in F₃() by A5; :: thesis: S₁[x,z]
let x2, y2 be object ; :: thesis: ( x = [x2,y2] implies P₁[x2,y2,z] )
assume A7: x = [x2,y2] ; :: thesis: P₁[x2,y2,z]
then x1 = x2 by A4, XTUPLE_0:1;
hence P₁[x2,y2,z] by A4, A6, A7, XTUPLE_0:1; :: thesis: verum

end;

consider f being Function of [:F₁(),F₂():],F₃() such that
A8: for x being object st x in [:F₁(),F₂():] holds
S₁[x,f . x] from FUNCT_2:sch 1(A2);
take f ; :: thesis: for x, y being object st x in F₁() & y in F₂() holds
P₁[x,y,f . (x,y)]
let x, y be object ; :: thesis: ( x in F₁() & y in F₂() implies P₁[x,y,f . (x,y)] )
assume ( x in F₁() & y in F₂() ) ; :: thesis: P₁[x,y,f . (x,y)]
then [x,y] in [:F₁(),F₂():] by ZFMISC_1:def 2;
hence P₁[x,y,f . (x,y)] by A8; :: thesis: verum