Step |
Hyp |
Ref |
Expression |
1 |
|
ovexd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 0 ..^ 𝑁 ) ∈ V ) |
2 |
|
elmapg |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ ( 0 ..^ 𝑁 ) ∈ V ) → ( 𝑤 ∈ ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) ↔ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) ) |
3 |
1 2
|
syldan |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑤 ∈ ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) ↔ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) ) |
4 |
|
iswrdi |
⊢ ( 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 → 𝑤 ∈ Word 𝑉 ) |
5 |
4
|
adantl |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) → 𝑤 ∈ Word 𝑉 ) |
6 |
|
fnfzo0hash |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) → ( ♯ ‘ 𝑤 ) = 𝑁 ) |
7 |
6
|
adantll |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) → ( ♯ ‘ 𝑤 ) = 𝑁 ) |
8 |
5 7
|
jca |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) → ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) ) |
9 |
8
|
ex |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 → ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) ) ) |
10 |
|
wrdf |
⊢ ( 𝑤 ∈ Word 𝑉 → 𝑤 : ( 0 ..^ ( ♯ ‘ 𝑤 ) ) ⟶ 𝑉 ) |
11 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝑤 ) = 𝑁 → ( 0 ..^ ( ♯ ‘ 𝑤 ) ) = ( 0 ..^ 𝑁 ) ) |
12 |
11
|
feq2d |
⊢ ( ( ♯ ‘ 𝑤 ) = 𝑁 → ( 𝑤 : ( 0 ..^ ( ♯ ‘ 𝑤 ) ) ⟶ 𝑉 ↔ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) ) |
13 |
10 12
|
syl5ibcom |
⊢ ( 𝑤 ∈ Word 𝑉 → ( ( ♯ ‘ 𝑤 ) = 𝑁 → 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) ) |
14 |
13
|
imp |
⊢ ( ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) → 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) |
15 |
9 14
|
impbid1 |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ↔ ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) ) ) |
16 |
3 15
|
bitrd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑤 ∈ ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) ↔ ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) ) ) |
17 |
16
|
abbi2dv |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) = { 𝑤 ∣ ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) } ) |
18 |
|
df-rab |
⊢ { 𝑤 ∈ Word 𝑉 ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } = { 𝑤 ∣ ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) } |
19 |
17 18
|
syl6reqr |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → { 𝑤 ∈ Word 𝑉 ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } = ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) ) |