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Metamath Proof Explorer

Theorem eupth2lem3lem5

Description: Lemma for eupth2 . (Contributed by AV, 25-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 ) ) ) )
eupth2lem3.o ⊢ ( 𝜑 → { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) } = if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑁 ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 𝑁 ) } ) )
eupth2lem3.e ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } )
Assertion eupth2lem3lem5 ( 𝜑 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ 𝒫 𝑉 )

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 eupth2lem3.o ⊢ ( 𝜑 → { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝑋 ) ‘ 𝑥 ) } = if ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 𝑁 ) , ∅ , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 𝑁 ) } ) )
14 eupth2lem3.e ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) = { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } )
15 1 2 3 4 5 6 trlsegvdeglem1 ⊢ ( 𝜑 → ( ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 ∧ ( 𝑃 ‘ ( 𝑁 + 1 ) ) ∈ 𝑉 ) )
16 prelpwi ⊢ ( ( ( 𝑃 ‘ 𝑁 ) ∈ 𝑉 ∧ ( 𝑃 ‘ ( 𝑁 + 1 ) ) ∈ 𝑉 ) → { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ∈ 𝒫 𝑉 )
17 15 16 syl ⊢ ( 𝜑 → { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ∈ 𝒫 𝑉 )
18 14 17 eqeltrd ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ 𝒫 𝑉 )