Step |
Hyp |
Ref |
Expression |
1 |
|
rdgfun |
⊢ Fun rec ( ( 𝑥 ∈ V ↦ 𝒫 𝑥 ) , ∅ ) |
2 |
|
df-r1 |
⊢ 𝑅1 = rec ( ( 𝑥 ∈ V ↦ 𝒫 𝑥 ) , ∅ ) |
3 |
2
|
funeqi |
⊢ ( Fun 𝑅1 ↔ Fun rec ( ( 𝑥 ∈ V ↦ 𝒫 𝑥 ) , ∅ ) ) |
4 |
1 3
|
mpbir |
⊢ Fun 𝑅1 |
5 |
|
rdgdmlim |
⊢ Lim dom rec ( ( 𝑥 ∈ V ↦ 𝒫 𝑥 ) , ∅ ) |
6 |
2
|
dmeqi |
⊢ dom 𝑅1 = dom rec ( ( 𝑥 ∈ V ↦ 𝒫 𝑥 ) , ∅ ) |
7 |
|
limeq |
⊢ ( dom 𝑅1 = dom rec ( ( 𝑥 ∈ V ↦ 𝒫 𝑥 ) , ∅ ) → ( Lim dom 𝑅1 ↔ Lim dom rec ( ( 𝑥 ∈ V ↦ 𝒫 𝑥 ) , ∅ ) ) ) |
8 |
6 7
|
ax-mp |
⊢ ( Lim dom 𝑅1 ↔ Lim dom rec ( ( 𝑥 ∈ V ↦ 𝒫 𝑥 ) , ∅ ) ) |
9 |
5 8
|
mpbir |
⊢ Lim dom 𝑅1 |
10 |
4 9
|
pm3.2i |
⊢ ( Fun 𝑅1 ∧ Lim dom 𝑅1 ) |