Theorem edgumgr

Description: Properties of an edge of a multigraph. (Contributed by AV, 25-Nov-2020)

	Ref	Expression
Assertion	edgumgr	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐸 ) = 2 ) )

Proof

Step	Hyp	Ref	Expression
1		umgredgss	⊢ ( 𝐺 ∈ UMGraph → ( Edg ‘ 𝐺 ) ⊆ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
2	1	sselda	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → 𝐸 ∈ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
3		fveqeq2	⊢ ( 𝑥 = 𝐸 → ( ( ♯ ‘ 𝑥 ) = 2 ↔ ( ♯ ‘ 𝐸 ) = 2 ) )
4	3	elrab	⊢ ( 𝐸 ∈ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ↔ ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐸 ) = 2 ) )
5	2 4	sylib	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐸 ) = 2 ) )