Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
id |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 ∈ 𝐷 ) |
3 |
1
|
psrbagf |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 : 𝐼 ⟶ ℕ0 ) |
4 |
3
|
ffnd |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼 ) |
5 |
2 4
|
fndmexd |
⊢ ( 𝐹 ∈ 𝐷 → 𝐼 ∈ V ) |
6 |
1
|
psrbag |
⊢ ( 𝐼 ∈ V → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) ) |
7 |
6
|
biimpa |
⊢ ( ( 𝐼 ∈ V ∧ 𝐹 ∈ 𝐷 ) → ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) |
8 |
5 7
|
mpancom |
⊢ ( 𝐹 ∈ 𝐷 → ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) |
9 |
8
|
simprd |
⊢ ( 𝐹 ∈ 𝐷 → ( ◡ 𝐹 “ ℕ ) ∈ Fin ) |
10 |
|
frnnn0fsuppg |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 finSupp 0 ↔ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) |
11 |
3 10
|
mpdan |
⊢ ( 𝐹 ∈ 𝐷 → ( 𝐹 finSupp 0 ↔ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) |
12 |
9 11
|
mpbird |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 finSupp 0 ) |