Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
simpr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℕ0 ) |
3 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
4 |
3
|
a1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → 0 ∈ ℕ0 ) |
5 |
2 4
|
ifcld |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ∈ ℕ0 ) |
6 |
5
|
adantr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑦 ∈ 𝐼 ) → if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ∈ ℕ0 ) |
7 |
6
|
fmpttd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) : 𝐼 ⟶ ℕ0 ) |
8 |
|
id |
⊢ ( 𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉 ) |
9 |
|
c0ex |
⊢ 0 ∈ V |
10 |
9
|
a1i |
⊢ ( 𝐼 ∈ 𝑉 → 0 ∈ V ) |
11 |
|
eqid |
⊢ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) |
12 |
8 10 11
|
sniffsupp |
⊢ ( 𝐼 ∈ 𝑉 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) finSupp 0 ) |
13 |
12
|
adantr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) finSupp 0 ) |
14 |
|
frnnn0fsupp |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) : 𝐼 ⟶ ℕ0 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) finSupp 0 ↔ ( ◡ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) “ ℕ ) ∈ Fin ) ) |
15 |
14
|
adantlr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) : 𝐼 ⟶ ℕ0 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) finSupp 0 ↔ ( ◡ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) “ ℕ ) ∈ Fin ) ) |
16 |
15
|
bicomd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) : 𝐼 ⟶ ℕ0 ) → ( ( ◡ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) “ ℕ ) ∈ Fin ↔ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) finSupp 0 ) ) |
17 |
7 16
|
mpdan |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( ( ◡ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) “ ℕ ) ∈ Fin ↔ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) finSupp 0 ) ) |
18 |
13 17
|
mpbird |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( ◡ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) “ ℕ ) ∈ Fin ) |
19 |
1
|
psrbag |
⊢ ( 𝐼 ∈ 𝑉 → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) ∈ 𝐷 ↔ ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) : 𝐼 ⟶ ℕ0 ∧ ( ◡ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) “ ℕ ) ∈ Fin ) ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) ∈ 𝐷 ↔ ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) : 𝐼 ⟶ ℕ0 ∧ ( ◡ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) “ ℕ ) ∈ Fin ) ) ) |
21 |
7 18 20
|
mpbir2and |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 𝑁 , 0 ) ) ∈ 𝐷 ) |