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Metamath Proof Explorer

Theorem umgr2v2eiedg

Description: The edge function in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020)

Ref Expression
Hypothesis umgr2v2evtx.g ⊢ 𝐺 = ⟨ 𝑉 , { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } ⟩
Assertion umgr2v2eiedg ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( iEdg ‘ 𝐺 ) = { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } )

Proof

Step Hyp Ref Expression
1 umgr2v2evtx.g ⊢ 𝐺 = ⟨ 𝑉 , { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } ⟩
2 1 fveq2i ⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ ⟨ 𝑉 , { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } ⟩ )
3 simp1 ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝑉 ∈ 𝑊 )
4 prex ⊢ { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } ∈ V
5 opiedgfv ⊢ ( ( 𝑉 ∈ 𝑊 ∧ { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } ∈ V ) → ( iEdg ‘ ⟨ 𝑉 , { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } ⟩ ) = { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } )
6 3 4 5 sylancl ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( iEdg ‘ ⟨ 𝑉 , { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } ⟩ ) = { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } )
7 2 6 eqtrid ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( iEdg ‘ 𝐺 ) = { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } )