Step |
Hyp |
Ref |
Expression |
1 |
|
reprval.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℕ ) |
2 |
|
reprval.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
reprval.s |
⊢ ( 𝜑 → 𝑆 ∈ ℕ0 ) |
4 |
|
reprss.1 |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 ) |
5 |
|
nnex |
⊢ ℕ ∈ V |
6 |
5
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
7 |
6 1
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
8 |
|
mapss |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ⊆ 𝐴 ) → ( 𝐵 ↑m ( 0 ..^ 𝑆 ) ) ⊆ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ) |
9 |
7 4 8
|
syl2anc |
⊢ ( 𝜑 → ( 𝐵 ↑m ( 0 ..^ 𝑆 ) ) ⊆ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ) |
10 |
9
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐵 ↑m ( 0 ..^ 𝑆 ) ) ) → 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ) |
11 |
10
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ ( 𝐵 ↑m ( 0 ..^ 𝑆 ) ) ∧ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 ) ) → 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ) |
12 |
11
|
rabss3d |
⊢ ( 𝜑 → { 𝑐 ∈ ( 𝐵 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } ⊆ { 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } ) |
13 |
4 1
|
sstrd |
⊢ ( 𝜑 → 𝐵 ⊆ ℕ ) |
14 |
13 2 3
|
reprval |
⊢ ( 𝜑 → ( 𝐵 ( repr ‘ 𝑆 ) 𝑀 ) = { 𝑐 ∈ ( 𝐵 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } ) |
15 |
1 2 3
|
reprval |
⊢ ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) = { 𝑐 ∈ ( 𝐴 ↑m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐 ‘ 𝑎 ) = 𝑀 } ) |
16 |
12 14 15
|
3sstr4d |
⊢ ( 𝜑 → ( 𝐵 ( repr ‘ 𝑆 ) 𝑀 ) ⊆ ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ) |