Step |
Hyp |
Ref |
Expression |
1 |
|
ffun |
⊢ ( 𝐹 : 𝐼 ⟶ ℕ0 → Fun 𝐹 ) |
2 |
|
simpl |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) → 𝐹 ∈ 𝑉 ) |
3 |
|
c0ex |
⊢ 0 ∈ V |
4 |
|
funisfsupp |
⊢ ( ( Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ V ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 supp 0 ) ∈ Fin ) ) |
5 |
3 4
|
mp3an3 |
⊢ ( ( Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 supp 0 ) ∈ Fin ) ) |
6 |
1 2 5
|
syl2an2 |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 supp 0 ) ∈ Fin ) ) |
7 |
|
frnnn0suppg |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 supp 0 ) = ( ◡ 𝐹 “ ℕ ) ) |
8 |
7
|
eleq1d |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) → ( ( 𝐹 supp 0 ) ∈ Fin ↔ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) |
9 |
6 8
|
bitrd |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 finSupp 0 ↔ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) |