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 V being set such that
A2: for x being object holds
( x in V iff ( x in F1() & S2[x] ) ) from A3: for x being object st x in V holds
ex y being object st S1[x,y]
proof
let x be object ; :: thesis: ( x in V implies ex y being object st S1[x,y] )
assume A4: x in V ; :: thesis: ex y being object st S1[x,y]
then reconsider d9 = x as Element of F1() by A2;
now :: thesis: ex y being Element of F2() ex y being object st S1[x,y]
per cases ( P1[d9] or P2[d9] or P3[d9] ) by A2, A4;
suppose A5: P1[d9] ; :: thesis: ex y being Element of F2() ex y being object st S1[x,y]
take y = F3(d9); :: thesis: ex y being object st S1[x,y]
now :: thesis: ex y being object st S1[x,y]
per cases ( P2[d9] or not P2[d9] ) ;
suppose A6: P2[d9] ; :: thesis: ex y being object st S1[x,y]
then A7: F3(d9) = F4(d9) by A1, A5;
now :: thesis: ex y being object st S1[x,y]end;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
suppose A8: P2[d9] ; :: thesis: ex y being object st S1[x,y]
now :: thesis: ex y being object st S1[x,y]end;
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;
suppose A9: P2[d9] ; :: thesis: ex y being Element of F2() ex y being object st S1[x,y]
take y = F4(x); :: thesis: ex y being object st S1[x,y]
now :: thesis: ex y being object st S1[x,y]
per cases ( P1[d9] or not P1[d9] ) ;
suppose P1[d9] ; :: thesis: ex y being object st S1[x,y]
then A10: F3(d9) = F4(d9) by A1, A9;
now :: thesis: ex y being object st S1[x,y]end;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
suppose A11: P1[d9] ; :: thesis: ex y being object st S1[x,y]
now :: thesis: ex y being object st S1[x,y]end;
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;
suppose A12: P3[d9] ; :: thesis: ex y being Element of F2() ex y being object st S1[x,y]
take y = F5(x); :: thesis: ex y being object st S1[x,y]
now :: thesis: ex y being object st S1[x,y]
per cases ( P1[d9] or not P1[d9] ) ;
suppose P1[d9] ; :: thesis: ex y being object st S1[x,y]
then A13: F3(d9) = F5(d9) by ;
now :: thesis: ex y being object st S1[x,y]end;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
suppose A14: P1[d9] ; :: thesis: ex y being object st S1[x,y]
now :: thesis: ex y being object st S1[x,y]end;
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;
end;
end;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
consider f being Function such that
A15: ( dom f = V & ( for x being object st x in V holds
S1[x,f . x] ) ) from A16: 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
A17: y in dom f and
A18: x = f . y by FUNCT_1:def 3;
now :: thesis: x in F2()end;
hence x in F2() ; :: thesis: verum
end;
V c= F1() by A2;
then reconsider q = f as PartFunc of F1(),F2() by ;
take q ; :: thesis: ( ( for c being Element of F1() holds
( c in dom q iff ( P1[c] or P2[c] or P3[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) ) & ( P3[c] implies q . c = F5(c) ) ) ) )

thus for c being Element of F1() holds
( c in dom q iff ( P1[c] or P2[c] or P3[c] ) ) by ; :: 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) ) & ( P3[c] implies q . c = F5(c) ) )

let i be Element of F1(); :: thesis: ( i in dom q implies ( ( P1[i] implies q . i = F3(i) ) & ( P2[i] implies q . i = F4(i) ) & ( P3[i] implies q . i = F5(i) ) ) )
assume i in dom q ; :: thesis: ( ( P1[i] implies q . i = F3(i) ) & ( P2[i] implies q . i = F4(i) ) & ( P3[i] implies q . i = F5(i) ) )
hence ( ( P1[i] implies q . i = F3(i) ) & ( P2[i] implies q . i = F4(i) ) & ( P3[i] implies q . i = F5(i) ) ) by A15; :: thesis: verum