Step |
Hyp |
Ref |
Expression |
1 |
|
vtxdgf.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
vtxdg0e.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
vtxdgfisnn0.a |
⊢ 𝐴 = dom 𝐼 |
4 |
1
|
vtxdgf |
⊢ ( 𝐺 ∈ 𝑊 → ( VtxDeg ‘ 𝐺 ) : 𝑉 ⟶ ℕ0* ) |
5 |
4
|
adantr |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin ) → ( VtxDeg ‘ 𝐺 ) : 𝑉 ⟶ ℕ0* ) |
6 |
5
|
ffnd |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin ) → ( VtxDeg ‘ 𝐺 ) Fn 𝑉 ) |
7 |
1 2 3
|
vtxdgfisnn0 |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝑢 ∈ 𝑉 ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑢 ) ∈ ℕ0 ) |
8 |
7
|
adantll |
⊢ ( ( ( 𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin ) ∧ 𝑢 ∈ 𝑉 ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑢 ) ∈ ℕ0 ) |
9 |
8
|
ralrimiva |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin ) → ∀ 𝑢 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑢 ) ∈ ℕ0 ) |
10 |
|
ffnfv |
⊢ ( ( VtxDeg ‘ 𝐺 ) : 𝑉 ⟶ ℕ0 ↔ ( ( VtxDeg ‘ 𝐺 ) Fn 𝑉 ∧ ∀ 𝑢 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑢 ) ∈ ℕ0 ) ) |
11 |
6 9 10
|
sylanbrc |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin ) → ( VtxDeg ‘ 𝐺 ) : 𝑉 ⟶ ℕ0 ) |