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Metamath Proof Explorer

Theorem umgr2v2eedg

Description: The set of edges 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 umgr2v2eedg ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( Edg ‘ 𝐺 ) = { { 𝐴 , 𝐵 } } )

Proof

Step Hyp Ref Expression
1 umgr2v2evtx.g ⊢ 𝐺 = ⟨ 𝑉 , { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } ⟩
2 edgval ⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
3 2 a1i ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) )
4 1 umgr2v2eiedg ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( iEdg ‘ 𝐺 ) = { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } )
5 4 rneqd ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ran ( iEdg ‘ 𝐺 ) = ran { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } )
6 c0ex ⊢ 0 ∈ V
7 1ex ⊢ 1 ∈ V
8 rnpropg ⊢ ( ( 0 ∈ V ∧ 1 ∈ V ) → ran { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } = { { 𝐴 , 𝐵 } , { 𝐴 , 𝐵 } } )
9 6 7 8 mp2an ⊢ ran { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } = { { 𝐴 , 𝐵 } , { 𝐴 , 𝐵 } }
10 9 a1i ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ran { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } = { { 𝐴 , 𝐵 } , { 𝐴 , 𝐵 } } )
11 dfsn2 ⊢ { { 𝐴 , 𝐵 } } = { { 𝐴 , 𝐵 } , { 𝐴 , 𝐵 } }
12 10 11 eqtr4di ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ran { ⟨ 0 , { 𝐴 , 𝐵 } ⟩ , ⟨ 1 , { 𝐴 , 𝐵 } ⟩ } = { { 𝐴 , 𝐵 } } )
13 3 5 12 3eqtrd ⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( Edg ‘ 𝐺 ) = { { 𝐴 , 𝐵 } } )