Theorem nbedgusgr

Description: The number of neighbors of a vertex is the number of edges at the vertex in a simple graph. (Contributed by AV, 27-Dec-2020) (Proof shortened by AV, 5-May-2021)

	Ref	Expression
Hypotheses	nbusgrf1o.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	nbusgrf1o.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	nbedgusgr	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉 ) → ( ♯ ‘ ( 𝐺 NeighbVtx 𝑈 ) ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } ) )

Proof

Step	Hyp	Ref	Expression
1		nbusgrf1o.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		nbusgrf1o.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		ovex	⊢ ( 𝐺 NeighbVtx 𝑈 ) ∈ V
4	1 2	nbusgrf1o	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉 ) → ∃ 𝑓 𝑓 : ( 𝐺 NeighbVtx 𝑈 ) –1-1-onto→ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } )
5		hasheqf1oi	⊢ ( ( 𝐺 NeighbVtx 𝑈 ) ∈ V → ( ∃ 𝑓 𝑓 : ( 𝐺 NeighbVtx 𝑈 ) –1-1-onto→ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } → ( ♯ ‘ ( 𝐺 NeighbVtx 𝑈 ) ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } ) ) )
6	3 4 5	mpsyl	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉 ) → ( ♯ ‘ ( 𝐺 NeighbVtx 𝑈 ) ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } ) )