Theorem upgrfn

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Hypotheses	isupgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isupgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	upgrfn	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ) → 𝐸 : 𝐴 ⟶ { 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	upgrf	⊢ ( 𝐺 ∈ UPGraph → 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 ∈ UPGraph → ( 𝐸 Fn 𝐴 → 𝐸 : 𝐴 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
7	6	imp	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ) → 𝐸 : 𝐴 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )