defpred S₁[ object , object ] means ( ( P₁[$1] implies $2 = F₃($1) ) & ( P₂[$1] implies $2 = F₄($1) ) );
defpred S₂[ object ] means ( P₁[$1] or P₂[$1] );
consider A being set such that
A4: for x being object holds
( x in A iff ( x in F₁() & S₂[x] ) ) from XBOOLE_0:sch 1();
A5: for x being object st x in A holds
ex y being object st S₁[x,y]

proof

let x be object ; :: thesis: ( x in A implies ex y being object st S₁[x,y] )
assume A6: x in A ; :: thesis: ex y being object st S₁[x,y]
then A7: x in F₁() by A4;

now :: thesis: ex y being set ex y being object st S₁[x,y]

per cases ( P₁[x] or P₂[x] ) by A4, A6;

suppose A8: P₁[x] ; :: thesis: ex y being set ex y being object st S₁[x,y]

take y = F₃(x); :: thesis: ex y being object st S₁[x,y]
P₂[x] by A1, A7, A8;
hence ex y being object st S₁[x,y] ; :: thesis: verum

end;

suppose A9: P₂[x] ; :: thesis: ex y being set ex y being object st S₁[x,y]

take y = F₄(x); :: thesis: ex y being object st S₁[x,y]
P₁[x] by A1, A7, A9;
hence ex y being object st S₁[x,y] ; :: thesis: verum

end;

end;

end;

hence ex y being object st S₁[x,y] ; :: thesis: verum

end;

consider f being Function such that
A10: ( dom f = A & ( for x being object st x in A holds
S₁[x,f . x] ) ) from CLASSES1:sch 1(A5);
A11: A c= F₁() by A4;
rng f c= F₂()

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng f or x in F₂() )
assume x in rng f ; :: thesis: x in F₂()
then consider y being object such that
A12: y in dom f and
A13: x = f . y by FUNCT_1:def 3;

now :: thesis: x in F₂()

per cases ( P₁[y] or P₂[y] ) by A4, A10, A12;

suppose A14: P₁[y] ; :: thesis: x in F₂()

then f . y = F₃(y) by A10, A12;
hence x in F₂() by A2, A11, A10, A12, A13, A14; :: thesis: verum

end;

suppose A15: P₂[y] ; :: thesis: x in F₂()

then f . y = F₄(y) by A10, A12;
hence x in F₂() by A3, A11, A10, A12, A13, A15; :: thesis: verum

end;

end;

end;

hence x in F₂() ; :: thesis: verum

end;

then reconsider f = f as PartFunc of F₁(),F₂() by A11, A10, RELSET_1:4;
take f ; :: thesis: ( ( for x being object holds
( x in dom f iff ( x in F₁() & ( P₁[x] or P₂[x] ) ) ) ) & ( for x being object st x in dom f holds
( ( P₁[x] implies f . x = F₃(x) ) & ( P₂[x] implies f . x = F₄(x) ) ) ) )
thus for x being object holds
( x in dom f iff ( x in F₁() & ( P₁[x] or P₂[x] ) ) ) by A4, A10; :: thesis: for x being object st x in dom f holds
( ( P₁[x] implies f . x = F₃(x) ) & ( P₂[x] implies f . x = F₄(x) ) )
let x be object ; :: thesis: ( x in dom f implies ( ( P₁[x] implies f . x = F₃(x) ) & ( P₂[x] implies f . x = F₄(x) ) ) )
assume x in dom f ; :: thesis: ( ( P₁[x] implies f . x = F₃(x) ) & ( P₂[x] implies f . x = F₄(x) ) )
hence ( ( P₁[x] implies f . x = F₃(x) ) & ( P₂[x] implies f . x = F₄(x) ) ) by A10; :: thesis: verum