defpred S1[ object , object ] means ( ( P1[\$1] implies \$2 = F3(\$1) ) & ( P2[\$1] implies \$2 = F4(\$1) ) & ( P3[\$1] implies \$2 = F5(\$1) ) );
defpred S2[ object ] means ( P1[\$1] or P2[\$1] or P3[\$1] );
consider S being set such that
A5: for x being object holds
( x in S iff ( x in F1() & S2[x] ) ) from A6: for x being object st x in S holds
ex y being object st S1[x,y]
proof
let x be object ; :: thesis: ( x in S implies ex y being object st S1[x,y] )
assume A7: x in S ; :: thesis: ex y being object st S1[x,y]
then A8: x in F1() by A5;
now :: thesis: ex y being set ex y being object st S1[x,y]
per cases ( P1[x] or P2[x] or P3[x] ) by A5, A7;
suppose A9: 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] & P3[x] ) by A1, A8, A9;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
suppose A10: 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] & P3[x] ) by A1, A8, A10;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
suppose A11: P3[x] ; :: thesis: ex y being set ex y being object st S1[x,y]
take y = F5(x); :: thesis: ex y being object st S1[x,y]
( P1[x] & P2[x] ) by A1, A8, A11;
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
A12: ( dom f = S & ( for x being object st x in S holds
S1[x,f . x] ) ) from A13: S c= F1() by A5;
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
A14: y in dom f and
A15: x = f . y by FUNCT_1:def 3;
now :: thesis: x in F2()end;
hence x in F2() ; :: thesis: verum
end;
then reconsider f = f as PartFunc of F1(),F2() by ;
take f ; :: thesis: ( ( for x being object holds
( x in dom f iff ( x in F1() & ( P1[x] or P2[x] or P3[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) ) & ( P3[x] implies f . x = F5(x) ) ) ) )

thus for x being object holds
( x in dom f iff ( x in F1() & ( P1[x] or P2[x] or P3[x] ) ) ) by ; :: 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) ) & ( P3[x] implies f . x = F5(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) ) & ( P3[x] implies f . x = F5(x) ) ) )
assume x in dom f ; :: thesis: ( ( P1[x] implies f . x = F3(x) ) & ( P2[x] implies f . x = F4(x) ) & ( P3[x] implies f . x = F5(x) ) )
hence ( ( P1[x] implies f . x = F3(x) ) & ( P2[x] implies f . x = F4(x) ) & ( P3[x] implies f . x = F5(x) ) ) by A12; :: thesis: verum