Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
⊢ ( 𝑆 ∈ 𝑉 → 𝑆 ∈ V ) |
2 |
1
|
adantr |
⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → 𝑆 ∈ V ) |
3 |
|
simpr |
⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℕ0 ) |
4 |
|
ovex |
⊢ ( 0 ..^ 𝑁 ) ∈ V |
5 |
|
mptexg |
⊢ ( ( 0 ..^ 𝑁 ) ∈ V → ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ 𝑆 ) ∈ V ) |
6 |
4 5
|
mp1i |
⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ 𝑆 ) ∈ V ) |
7 |
|
oveq2 |
⊢ ( 𝑛 = 𝑁 → ( 0 ..^ 𝑛 ) = ( 0 ..^ 𝑁 ) ) |
8 |
7
|
adantl |
⊢ ( ( 𝑠 = 𝑆 ∧ 𝑛 = 𝑁 ) → ( 0 ..^ 𝑛 ) = ( 0 ..^ 𝑁 ) ) |
9 |
|
simpll |
⊢ ( ( ( 𝑠 = 𝑆 ∧ 𝑛 = 𝑁 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑛 ) ) → 𝑠 = 𝑆 ) |
10 |
8 9
|
mpteq12dva |
⊢ ( ( 𝑠 = 𝑆 ∧ 𝑛 = 𝑁 ) → ( 𝑥 ∈ ( 0 ..^ 𝑛 ) ↦ 𝑠 ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ 𝑆 ) ) |
11 |
|
df-reps |
⊢ repeatS = ( 𝑠 ∈ V , 𝑛 ∈ ℕ0 ↦ ( 𝑥 ∈ ( 0 ..^ 𝑛 ) ↦ 𝑠 ) ) |
12 |
10 11
|
ovmpoga |
⊢ ( ( 𝑆 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ 𝑆 ) ∈ V ) → ( 𝑆 repeatS 𝑁 ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ 𝑆 ) ) |
13 |
2 3 6 12
|
syl3anc |
⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑆 repeatS 𝑁 ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ 𝑆 ) ) |