thus { st1 where st1 is Element of F3() : ( st1 in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } & P2[st1] ) } c= { F4(s2,t2) where s2 is Element of F1(), t2 is Element of F2() : ( P1[s2,t2] & P2[F4(s2,t2)] ) } :: according to XBOOLE_0:def 10 :: thesis: { F4(s2,t2) where s2 is Element of F1(), t2 is Element of F2() : ( P1[s2,t2] & P2[F4(s2,t2)] ) } c= { st1 where st1 is Element of F3() : ( st1 in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } & P2[st1] ) }
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { st1 where st1 is Element of F3() : ( st1 in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } & P2[st1] ) } or x in { F4(s2,t2) where s2 is Element of F1(), t2 is Element of F2() : ( P1[s2,t2] & P2[F4(s2,t2)] ) } )
assume x in { st1 where st1 is Element of F3() : ( st1 in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } & P2[st1] ) } ; :: thesis: x in { F4(s2,t2) where s2 is Element of F1(), t2 is Element of F2() : ( P1[s2,t2] & P2[F4(s2,t2)] ) }
then consider st1 being Element of F3() such that
A1: x = st1 and
A2: st1 in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } and
A3: P2[st1] ;
ex s1 being Element of F1() ex t1 being Element of F2() st
( st1 = F4(s1,t1) & P1[s1,t1] ) by A2;
hence x in { F4(s2,t2) where s2 is Element of F1(), t2 is Element of F2() : ( P1[s2,t2] & P2[F4(s2,t2)] ) } by A1, A3; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { F4(s2,t2) where s2 is Element of F1(), t2 is Element of F2() : ( P1[s2,t2] & P2[F4(s2,t2)] ) } or x in { st1 where st1 is Element of F3() : ( st1 in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } & P2[st1] ) } )
assume x in { F4(s2,t2) where s2 is Element of F1(), t2 is Element of F2() : ( P1[s2,t2] & P2[F4(s2,t2)] ) } ; :: thesis: x in { st1 where st1 is Element of F3() : ( st1 in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } & P2[st1] ) }
then consider s2 being Element of F1(), t2 being Element of F2() such that
A4: x = F4(s2,t2) and
A5: P1[s2,t2] and
A6: P2[F4(s2,t2)] ;
F4(s2,t2) in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } by A5;
hence x in { st1 where st1 is Element of F3() : ( st1 in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } & P2[st1] ) } by A4, A6; :: thesis: verum