| Step |
Hyp |
Ref |
Expression |
| 1 |
|
stgrvtx0.g |
⊢ 𝐺 = ( StarGr ‘ 𝑁 ) |
| 2 |
|
stgrvtx0.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
| 3 |
|
stgrvtx |
⊢ ( 𝑁 ∈ ℕ0 → ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) ) |
| 4 |
1
|
fveq2i |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) |
| 5 |
2 4
|
eqtri |
⊢ 𝑉 = ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) |
| 6 |
5
|
eqeq1i |
⊢ ( 𝑉 = ( 0 ... 𝑁 ) ↔ ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) ) |
| 7 |
|
0elfz |
⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑁 ) ) |
| 8 |
|
eleq2 |
⊢ ( 𝑉 = ( 0 ... 𝑁 ) → ( 0 ∈ 𝑉 ↔ 0 ∈ ( 0 ... 𝑁 ) ) ) |
| 9 |
7 8
|
syl5ibrcom |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑉 = ( 0 ... 𝑁 ) → 0 ∈ 𝑉 ) ) |
| 10 |
6 9
|
biimtrrid |
⊢ ( 𝑁 ∈ ℕ0 → ( ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) → 0 ∈ 𝑉 ) ) |
| 11 |
3 10
|
mpd |
⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ 𝑉 ) |