Step |
Hyp |
Ref |
Expression |
1 |
|
mbfneg.1 |
|- ( ( ph /\ x e. A ) -> B e. V ) |
2 |
|
mbfneg.2 |
|- ( ph -> ( x e. A |-> B ) e. MblFn ) |
3 |
|
eqid |
|- ( x e. A |-> B ) = ( x e. A |-> B ) |
4 |
3 1
|
dmmptd |
|- ( ph -> dom ( x e. A |-> B ) = A ) |
5 |
2
|
dmexd |
|- ( ph -> dom ( x e. A |-> B ) e. _V ) |
6 |
4 5
|
eqeltrrd |
|- ( ph -> A e. _V ) |
7 |
|
neg1rr |
|- -u 1 e. RR |
8 |
7
|
a1i |
|- ( ( ph /\ x e. A ) -> -u 1 e. RR ) |
9 |
|
fconstmpt |
|- ( A X. { -u 1 } ) = ( x e. A |-> -u 1 ) |
10 |
9
|
a1i |
|- ( ph -> ( A X. { -u 1 } ) = ( x e. A |-> -u 1 ) ) |
11 |
|
eqidd |
|- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) ) |
12 |
6 8 1 10 11
|
offval2 |
|- ( ph -> ( ( A X. { -u 1 } ) oF x. ( x e. A |-> B ) ) = ( x e. A |-> ( -u 1 x. B ) ) ) |
13 |
2 1
|
mbfmptcl |
|- ( ( ph /\ x e. A ) -> B e. CC ) |
14 |
13
|
mulm1d |
|- ( ( ph /\ x e. A ) -> ( -u 1 x. B ) = -u B ) |
15 |
14
|
mpteq2dva |
|- ( ph -> ( x e. A |-> ( -u 1 x. B ) ) = ( x e. A |-> -u B ) ) |
16 |
12 15
|
eqtrd |
|- ( ph -> ( ( A X. { -u 1 } ) oF x. ( x e. A |-> B ) ) = ( x e. A |-> -u B ) ) |
17 |
7
|
a1i |
|- ( ph -> -u 1 e. RR ) |
18 |
13
|
fmpttd |
|- ( ph -> ( x e. A |-> B ) : A --> CC ) |
19 |
2 17 18
|
mbfmulc2re |
|- ( ph -> ( ( A X. { -u 1 } ) oF x. ( x e. A |-> B ) ) e. MblFn ) |
20 |
16 19
|
eqeltrrd |
|- ( ph -> ( x e. A |-> -u B ) e. MblFn ) |