Step |
Hyp |
Ref |
Expression |
1 |
|
n0i |
|- ( f e. ( A ^pm B ) -> -. ( A ^pm B ) = (/) ) |
2 |
|
fnpm |
|- ^pm Fn ( _V X. _V ) |
3 |
2
|
fndmi |
|- dom ^pm = ( _V X. _V ) |
4 |
3
|
ndmov |
|- ( -. ( A e. _V /\ B e. _V ) -> ( A ^pm B ) = (/) ) |
5 |
1 4
|
nsyl2 |
|- ( f e. ( A ^pm B ) -> ( A e. _V /\ B e. _V ) ) |
6 |
|
elpmg |
|- ( ( A e. _V /\ B e. _V ) -> ( f e. ( A ^pm B ) <-> ( Fun f /\ f C_ ( B X. A ) ) ) ) |
7 |
5 6
|
syl |
|- ( f e. ( A ^pm B ) -> ( f e. ( A ^pm B ) <-> ( Fun f /\ f C_ ( B X. A ) ) ) ) |
8 |
7
|
ibi |
|- ( f e. ( A ^pm B ) -> ( Fun f /\ f C_ ( B X. A ) ) ) |
9 |
8
|
simprd |
|- ( f e. ( A ^pm B ) -> f C_ ( B X. A ) ) |
10 |
|
velpw |
|- ( f e. ~P ( B X. A ) <-> f C_ ( B X. A ) ) |
11 |
9 10
|
sylibr |
|- ( f e. ( A ^pm B ) -> f e. ~P ( B X. A ) ) |
12 |
11
|
ssriv |
|- ( A ^pm B ) C_ ~P ( B X. A ) |