Metamath Proof Explorer


Theorem trlsegvdeglem6

Description: Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021)

Ref Expression
Hypotheses trlsegvdeg.v 𝑉 = ( Vtx ‘ 𝐺 )
trlsegvdeg.i 𝐼 = ( iEdg ‘ 𝐺 )
trlsegvdeg.f ( 𝜑 → Fun 𝐼 )
trlsegvdeg.n ( 𝜑𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
trlsegvdeg.u ( 𝜑𝑈𝑉 )
trlsegvdeg.w ( 𝜑𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
trlsegvdeg.vx ( 𝜑 → ( Vtx ‘ 𝑋 ) = 𝑉 )
trlsegvdeg.vy ( 𝜑 → ( Vtx ‘ 𝑌 ) = 𝑉 )
trlsegvdeg.vz ( 𝜑 → ( Vtx ‘ 𝑍 ) = 𝑉 )
trlsegvdeg.ix ( 𝜑 → ( iEdg ‘ 𝑋 ) = ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) )
trlsegvdeg.iy ( 𝜑 → ( iEdg ‘ 𝑌 ) = { ⟨ ( 𝐹𝑁 ) , ( 𝐼 ‘ ( 𝐹𝑁 ) ) ⟩ } )
trlsegvdeg.iz ( 𝜑 → ( iEdg ‘ 𝑍 ) = ( 𝐼 ↾ ( 𝐹 “ ( 0 ... 𝑁 ) ) ) )
Assertion trlsegvdeglem6 ( 𝜑 → dom ( iEdg ‘ 𝑋 ) ∈ Fin )

Proof

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 ( 𝜑𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
5 trlsegvdeg.u ( 𝜑𝑈𝑉 )
6 trlsegvdeg.w ( 𝜑𝐹 ( 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 ) = { ⟨ ( 𝐹𝑁 ) , ( 𝐼 ‘ ( 𝐹𝑁 ) ) ⟩ } )
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 ) 𝑃𝐹 : ( 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 )