Step |
Hyp |
Ref |
Expression |
1 |
|
reprval.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℕ ) |
2 |
|
reprval.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
reprval.s |
⊢ ( 𝜑 → 𝑆 ∈ ℕ0 ) |
4 |
|
reprfi.1 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
5 |
1 2 3
|
reprval |
⊢ ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) = { 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } ) |
6 |
|
fzofi |
⊢ ( 0 ..^ 𝑆 ) ∈ Fin |
7 |
|
mapfi |
⊢ ( ( 𝐴 ∈ Fin ∧ ( 0 ..^ 𝑆 ) ∈ Fin ) → ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∈ Fin ) |
8 |
4 6 7
|
sylancl |
⊢ ( 𝜑 → ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∈ Fin ) |
9 |
|
rabfi |
⊢ ( ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∈ Fin → { 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } ∈ Fin ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → { 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } ∈ Fin ) |
11 |
5 10
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ∈ Fin ) |