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 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑁 ) ∈ V |
14 |
|
fvex |
⊢ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ V |
15 |
13 14
|
pm3.2i |
⊢ ( ( 𝐹 ‘ 𝑁 ) ∈ V ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ V ) |
16 |
|
funsng |
⊢ ( ( ( 𝐹 ‘ 𝑁 ) ∈ V ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ V ) → Fun { 〈 ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) 〉 } ) |
17 |
15 16
|
mp1i |
⊢ ( 𝜑 → Fun { 〈 ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) 〉 } ) |
18 |
11
|
funeqd |
⊢ ( 𝜑 → ( Fun ( iEdg ‘ 𝑌 ) ↔ Fun { 〈 ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) 〉 } ) ) |
19 |
17 18
|
mpbird |
⊢ ( 𝜑 → Fun ( iEdg ‘ 𝑌 ) ) |