Step |
Hyp |
Ref |
Expression |
1 |
|
reprval.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℕ ) |
2 |
|
reprval.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
reprval.s |
⊢ ( 𝜑 → 𝑆 ∈ ℕ0 ) |
4 |
|
reprf.c |
⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ) |
5 |
1 2 3
|
reprval |
⊢ ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) = { 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } ) |
6 |
4 5
|
eleqtrd |
⊢ ( 𝜑 → 𝐶 ∈ { 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } ) |
7 |
|
elrabi |
⊢ ( 𝐶 ∈ { 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } → 𝐶 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ) |
8 |
|
elmapi |
⊢ ( 𝐶 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) → 𝐶 : ( 0 ..^ 𝑆 ) ⟶ 𝐴 ) |
9 |
6 7 8
|
3syl |
⊢ ( 𝜑 → 𝐶 : ( 0 ..^ 𝑆 ) ⟶ 𝐴 ) |