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Metamath Proof Explorer

Theorem usgredg2

Description: The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017) (Revised by AV, 16-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

		Ref	Expression
	Hypothesis	usgredg2.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
	Assertion	usgredg2	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( ♯ ‘ ( 𝐸 ‘ 𝑋 ) ) = 2 )

Proof

Step	Hyp	Ref	Expression
1		usgredg2.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2		usgrumgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
3		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
4	3 1	umgredg2	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( ♯ ‘ ( 𝐸 ‘ 𝑋 ) ) = 2 )
5	2 4	sylan	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( ♯ ‘ ( 𝐸 ‘ 𝑋 ) ) = 2 )