Step |
Hyp |
Ref |
Expression |
1 |
|
df-rab |
⊢ { 𝑤 ∈ Word 𝑉 ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } = { 𝑤 ∣ ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) } |
2 |
|
ovexd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 0 ..^ 𝑁 ) ∈ V ) |
3 |
|
elmapg |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ ( 0 ..^ 𝑁 ) ∈ V ) → ( 𝑤 ∈ ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) ↔ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) ) |
4 |
2 3
|
syldan |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑤 ∈ ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) ↔ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) ) |
5 |
|
iswrdi |
⊢ ( 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 → 𝑤 ∈ Word 𝑉 ) |
6 |
5
|
adantl |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) → 𝑤 ∈ Word 𝑉 ) |
7 |
|
fnfzo0hash |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) → ( ♯ ‘ 𝑤 ) = 𝑁 ) |
8 |
7
|
adantll |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) → ( ♯ ‘ 𝑤 ) = 𝑁 ) |
9 |
6 8
|
jca |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) → ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) ) |
10 |
9
|
ex |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 → ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) ) ) |
11 |
|
wrdf |
⊢ ( 𝑤 ∈ Word 𝑉 → 𝑤 : ( 0 ..^ ( ♯ ‘ 𝑤 ) ) ⟶ 𝑉 ) |
12 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝑤 ) = 𝑁 → ( 0 ..^ ( ♯ ‘ 𝑤 ) ) = ( 0 ..^ 𝑁 ) ) |
13 |
12
|
feq2d |
⊢ ( ( ♯ ‘ 𝑤 ) = 𝑁 → ( 𝑤 : ( 0 ..^ ( ♯ ‘ 𝑤 ) ) ⟶ 𝑉 ↔ 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) ) |
14 |
11 13
|
syl5ibcom |
⊢ ( 𝑤 ∈ Word 𝑉 → ( ( ♯ ‘ 𝑤 ) = 𝑁 → 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) ) |
15 |
14
|
imp |
⊢ ( ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) → 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ) |
16 |
10 15
|
impbid1 |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑤 : ( 0 ..^ 𝑁 ) ⟶ 𝑉 ↔ ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) ) ) |
17 |
4 16
|
bitrd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑤 ∈ ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) ↔ ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) ) ) |
18 |
17
|
abbi2dv |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) = { 𝑤 ∣ ( 𝑤 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑤 ) = 𝑁 ) } ) |
19 |
1 18
|
eqtr4id |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0 ) → { 𝑤 ∈ Word 𝑉 ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } = ( 𝑉 ↑m ( 0 ..^ 𝑁 ) ) ) |