Step |
Hyp |
Ref |
Expression |
1 |
|
rdgfun |
|- Fun rec ( ( x e. _V |-> ~P x ) , (/) ) |
2 |
|
df-r1 |
|- R1 = rec ( ( x e. _V |-> ~P x ) , (/) ) |
3 |
2
|
funeqi |
|- ( Fun R1 <-> Fun rec ( ( x e. _V |-> ~P x ) , (/) ) ) |
4 |
1 3
|
mpbir |
|- Fun R1 |
5 |
|
rdgdmlim |
|- Lim dom rec ( ( x e. _V |-> ~P x ) , (/) ) |
6 |
2
|
dmeqi |
|- dom R1 = dom rec ( ( x e. _V |-> ~P x ) , (/) ) |
7 |
|
limeq |
|- ( dom R1 = dom rec ( ( x e. _V |-> ~P x ) , (/) ) -> ( Lim dom R1 <-> Lim dom rec ( ( x e. _V |-> ~P x ) , (/) ) ) ) |
8 |
6 7
|
ax-mp |
|- ( Lim dom R1 <-> Lim dom rec ( ( x e. _V |-> ~P x ) , (/) ) ) |
9 |
5 8
|
mpbir |
|- Lim dom R1 |
10 |
4 9
|
pm3.2i |
|- ( Fun R1 /\ Lim dom R1 ) |