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Metamath Proof Explorer

Theorem upgrf

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Hypotheses	isupgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isupgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	upgrf	⊢ ( 𝐺 ∈ UPGraph → 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )

Proof

Step	Hyp	Ref	Expression
1		isupgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isupgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
3	1 2	isupgr	⊢ ( 𝐺 ∈ UPGraph → ( 𝐺 ∈ UPGraph ↔ 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
4	3	ibi	⊢ ( 𝐺 ∈ UPGraph → 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )