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Metamath Proof Explorer

Theorem trlsegvdeglem3

Description: Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021)

Ref Expression
Hypotheses trlsegvdeg.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
trlsegvdeg.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
trlsegvdeg.f ⊢ ( 𝜑 → Fun 𝐼 )
trlsegvdeg.n ⊢ ( 𝜑 → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
trlsegvdeg.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
trlsegvdeg.w ⊢ ( 𝜑 → 𝐹 ( 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 ) = { ⟨ ( 𝐹 ‘ 𝑁 ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ⟩ } )
trlsegvdeg.iz ⊢ ( 𝜑 → ( iEdg ‘ 𝑍 ) = ( 𝐼 ↾ ( 𝐹 “ ( 0 ... 𝑁 ) ) ) )
Assertion trlsegvdeglem3 ( 𝜑 → Fun ( iEdg ‘ 𝑌 ) )

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 ⊢ ( 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 ) )