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Metamath Proof Explorer

Theorem eupth2lem3lem1

Description: Lemma for eupth2lem3 . (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 ⊢ ( 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 eupth2lem3lem1 ( 𝜑 → ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ∈ ℕ0 )

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 5 7 eleqtrrd ⊢ ( 𝜑 → 𝑈 ∈ ( Vtx ‘ 𝑋 ) )
14 13 elfvexd ⊢ ( 𝜑 → 𝑋 ∈ V )
15 1 2 3 4 5 6 7 8 9 10 11 12 trlsegvdeglem6 ⊢ ( 𝜑 → dom ( iEdg ‘ 𝑋 ) ∈ Fin )
16 eqid ⊢ ( Vtx ‘ 𝑋 ) = ( Vtx ‘ 𝑋 )
17 eqid ⊢ ( iEdg ‘ 𝑋 ) = ( iEdg ‘ 𝑋 )
18 eqid ⊢ dom ( iEdg ‘ 𝑋 ) = dom ( iEdg ‘ 𝑋 )
19 16 17 18 vtxdgfisf ⊢ ( ( 𝑋 ∈ V ∧ dom ( iEdg ‘ 𝑋 ) ∈ Fin ) → ( VtxDeg ‘ 𝑋 ) : ( Vtx ‘ 𝑋 ) ⟶ ℕ0 )
20 14 15 19 syl2anc ⊢ ( 𝜑 → ( VtxDeg ‘ 𝑋 ) : ( Vtx ‘ 𝑋 ) ⟶ ℕ0 )
21 20 13 ffvelcdmd ⊢ ( 𝜑 → ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑈 ) ∈ ℕ0 )