Theorem umgrfn

Description: The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by AV, 24-Nov-2020)

	Ref	Expression
Hypotheses	isumgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isumgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	umgrfn	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐸 Fn 𝐴 ) → 𝐸 : 𝐴 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } )

Proof

Step	Hyp	Ref	Expression
1		isumgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isumgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
3	1 2	umgrf	⊢ ( 𝐺 ∈ UMGraph → 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } )
4		fndm	⊢ ( 𝐸 Fn 𝐴 → dom 𝐸 = 𝐴 )
5	4	feq2d	⊢ ( 𝐸 Fn 𝐴 → ( 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ↔ 𝐸 : 𝐴 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
6	3 5	syl5ibcom	⊢ ( 𝐺 ∈ UMGraph → ( 𝐸 Fn 𝐴 → 𝐸 : 𝐴 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
7	6	imp	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝐸 Fn 𝐴 ) → 𝐸 : 𝐴 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } )