Step |
Hyp |
Ref |
Expression |
1 |
|
fargshift.g |
⊢ 𝐺 = ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) |
2 |
|
ffn |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 → 𝐹 Fn ( 1 ... 𝑁 ) ) |
3 |
|
fseq1hash |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 Fn ( 1 ... 𝑁 ) ) → ( ♯ ‘ 𝐹 ) = 𝑁 ) |
4 |
2 3
|
sylan2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → ( ♯ ‘ 𝐹 ) = 𝑁 ) |
5 |
|
eleq1 |
⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( 𝑁 ∈ ℕ0 ↔ ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) ) |
6 |
|
oveq2 |
⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( 1 ... 𝑁 ) = ( 1 ... ( ♯ ‘ 𝐹 ) ) ) |
7 |
6
|
feq2d |
⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ↔ 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) ) |
8 |
5 7
|
anbi12d |
⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) ↔ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) ) ) |
9 |
8
|
eqcoms |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) ↔ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) ) ) |
10 |
|
fz0add1fz1 |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑥 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) |
11 |
|
ffvelrn |
⊢ ( ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ( 𝑥 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ ( 𝑥 + 1 ) ) ∈ dom 𝐸 ) |
12 |
11
|
expcom |
⊢ ( ( 𝑥 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝐹 ) ) → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( 𝐹 ‘ ( 𝑥 + 1 ) ) ∈ dom 𝐸 ) ) |
13 |
10 12
|
syl |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 → ( 𝐹 ‘ ( 𝑥 + 1 ) ) ∈ dom 𝐸 ) ) |
14 |
13
|
impancom |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) → ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐹 ‘ ( 𝑥 + 1 ) ) ∈ dom 𝐸 ) ) |
15 |
14
|
ralrimiv |
⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝐹 : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) → ∀ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ ( 𝑥 + 1 ) ) ∈ dom 𝐸 ) |
16 |
9 15
|
syl6bi |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → ∀ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ ( 𝑥 + 1 ) ) ∈ dom 𝐸 ) ) |
17 |
4 16
|
mpcom |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → ∀ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ ( 𝑥 + 1 ) ) ∈ dom 𝐸 ) |
18 |
1
|
fmpt |
⊢ ( ∀ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ ( 𝑥 + 1 ) ) ∈ dom 𝐸 ↔ 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) |
19 |
17 18
|
sylib |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) |