Step |
Hyp |
Ref |
Expression |
1 |
|
trlsegvdeg.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
trlsegvdeg.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
trlsegvdeg.f |
⊢ ( 𝜑 → Fun 𝐼 ) |
4 |
|
trlsegvdeg.n |
⊢ ( 𝜑 → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
5 |
|
trlsegvdeg.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
6 |
|
trlsegvdeg.w |
⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) |
7 |
|
trlsegvdeg.vx |
⊢ ( 𝜑 → ( Vtx ‘ 𝑋 ) = 𝑉 ) |
8 |
|
trlsegvdeg.vy |
⊢ ( 𝜑 → ( Vtx ‘ 𝑌 ) = 𝑉 ) |
9 |
|
trlsegvdeg.vz |
⊢ ( 𝜑 → ( Vtx ‘ 𝑍 ) = 𝑉 ) |
10 |
|
trlsegvdeg.ix |
⊢ ( 𝜑 → ( iEdg ‘ 𝑋 ) = ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) ) |
11 |
|
trlsegvdeg.iy |
⊢ ( 𝜑 → ( iEdg ‘ 𝑌 ) = { 〈 ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) 〉 } ) |
12 |
|
trlsegvdeg.iz |
⊢ ( 𝜑 → ( iEdg ‘ 𝑍 ) = ( 𝐼 ↾ ( 𝐹 “ ( 0 ... 𝑁 ) ) ) ) |
13 |
1 2 3 4 5 6 7 8 9 10 11 12
|
trlsegvdeglem4 |
⊢ ( 𝜑 → dom ( iEdg ‘ 𝑋 ) = ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) ) |
14 |
2
|
trlf1 |
⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ) |
15 |
|
f1fun |
⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 → Fun 𝐹 ) |
16 |
6 14 15
|
3syl |
⊢ ( 𝜑 → Fun 𝐹 ) |
17 |
|
fzofi |
⊢ ( 0 ..^ 𝑁 ) ∈ Fin |
18 |
|
imafi |
⊢ ( ( Fun 𝐹 ∧ ( 0 ..^ 𝑁 ) ∈ Fin ) → ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∈ Fin ) |
19 |
16 17 18
|
sylancl |
⊢ ( 𝜑 → ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∈ Fin ) |
20 |
|
infi |
⊢ ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∈ Fin → ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) ∈ Fin ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ∩ dom 𝐼 ) ∈ Fin ) |
22 |
13 21
|
eqeltrd |
⊢ ( 𝜑 → dom ( iEdg ‘ 𝑋 ) ∈ Fin ) |