defpred S₁[ set , set ] means ( ( P₁[$1] implies $2 = F₃($1) ) & ( P₂[$1] implies $2 = F₄($1) ) );
defpred S₂[ set ] means ( P₁[$1] or P₂[$1] );
consider Y being set such that
A2: for x being set holds
( x in Y iff ( x in F₁() & S₂[x] ) ) from XBOOLE_0:sch 1();
A3: for x being set st x in Y holds
ex y being set st S₁[x,y]

proof

let x be set ; :: thesis: ( x in Y implies ex y being set st S₁[x,y] )
assume A4: x in Y ; :: thesis: ex y being set st S₁[x,y]
then reconsider a9 = x as Element of F₁() by A2;

now

per cases ( P₁[a9] or P₂[a9] ) by A2, A4;

suppose A5: P₁[a9] ; :: thesis: ex y being Element of F₂() ex y being set st S₁[x,y]

take y = F₃(a9); :: thesis: ex y being set st S₁[x,y]

now

per cases ( P₂[a9] or not P₂[a9] ) ;

suppose P₂[a9] ; :: thesis: ex y being set st S₁[x,y]

then F₃(a9) = F₄(a9) by A1, A5;
hence ex y being set st S₁[x,y] ; :: thesis: verum

end;

suppose P₂[a9] ; :: thesis: ex y being set st S₁[x,y]

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

end;

end;

end;

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

end;

suppose A6: P₂[a9] ; :: thesis: ex y being Element of F₂() ex y being set st S₁[x,y]

take y = F₄(a9); :: thesis: ex y being set st S₁[x,y]

now

per cases ( P₁[a9] or not P₁[a9] ) ;

suppose P₁[a9] ; :: thesis: ex y being set st S₁[x,y]

then F₃(a9) = F₄(a9) by A1, A6;
hence ex y being set st S₁[x,y] ; :: thesis: verum

end;

suppose P₁[a9] ; :: thesis: ex y being set st S₁[x,y]

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

end;

end;

end;

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

end;

end;

end;

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

end;

consider f being Function such that
A7: ( dom f = Y & ( for x being set st x in Y holds
S₁[x,f . x] ) ) from CLASSES1:sch 1(A3);
A8: Y c= F₁()

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Y or x in F₁() )
assume x in Y ; :: thesis: x in F₁()
hence x in F₁() by A2; :: thesis: verum

end;

rng f c= F₂()

proof

let x be set ; :: 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 set such that
A9: y in dom f and
A10: x = f . y by FUNCT_1:def 3;

now

per cases ( P₁[y] or P₂[y] ) by A2, A7, A9;

suppose P₁[y] ; :: thesis: x in F₂()

then f . y = F₃(y) by A7, A9;
hence x in F₂() by A10; :: thesis: verum

end;

suppose P₂[y] ; :: thesis: x in F₂()

then f . y = F₄(y) by A7, A9;
hence x in F₂() by A10; :: thesis: verum

end;

end;

end;

hence x in F₂() ; :: thesis: verum

end;

then reconsider q = f as PartFunc of F₁(),F₂() by A8, A7, RELSET_1:4;
take q ; :: thesis: ( ( for c being Element of F₁() holds
( c in dom q iff ( P₁[c] or P₂[c] ) ) ) & ( for c being Element of F₁() st c in dom q holds
( ( P₁[c] implies q . c = F₃(c) ) & ( P₂[c] implies q . c = F₄(c) ) ) ) )
thus for c being Element of F₁() holds
( c in dom q iff ( P₁[c] or P₂[c] ) ) by A2, A7; :: thesis: for c being Element of F₁() st c in dom q holds
( ( P₁[c] implies q . c = F₃(c) ) & ( P₂[c] implies q . c = F₄(c) ) )
let e be Element of F₁(); :: thesis: ( e in dom q implies ( ( P₁[e] implies q . e = F₃(e) ) & ( P₂[e] implies q . e = F₄(e) ) ) )
assume e in dom q ; :: thesis: ( ( P₁[e] implies q . e = F₃(e) ) & ( P₂[e] implies q . e = F₄(e) ) )
hence ( ( P₁[e] implies q . e = F₃(e) ) & ( P₂[e] implies q . e = F₄(e) ) ) by A7; :: thesis: verum