Step |
Hyp |
Ref |
Expression |
1 |
|
wrdfn |
⊢ ( 𝑊 ∈ Word V → 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
2 |
1
|
anim1i |
⊢ ( ( 𝑊 ∈ Word V ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) ∈ 𝑉 ) → ( 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) ∈ 𝑉 ) ) |
3 |
|
ffnfv |
⊢ ( 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ↔ ( 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) ∈ 𝑉 ) ) |
4 |
2 3
|
sylibr |
⊢ ( ( 𝑊 ∈ Word V ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) ∈ 𝑉 ) → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ) |
5 |
|
iswrdi |
⊢ ( 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 → 𝑊 ∈ Word 𝑉 ) |
6 |
4 5
|
syl |
⊢ ( ( 𝑊 ∈ Word V ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) ∈ 𝑉 ) → 𝑊 ∈ Word 𝑉 ) |