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Metamath Proof Explorer

Theorem vtxdusgrval

Description: The value of the vertex degree function for a simple graph. (Contributed by Alexander van der Vekens, 20-Dec-2017) (Revised by AV, 11-Dec-2020)

		Ref	Expression
	Hypotheses	vtxdlfgrval.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
		vtxdlfgrval.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
		vtxdlfgrval.a	⊢ 𝐴 = dom 𝐼
		vtxdlfgrval.d	⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
	Assertion	vtxdusgrval	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝐷 ‘ 𝑈 ) = ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) )

Proof

Step	Hyp	Ref	Expression
1		vtxdlfgrval.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		vtxdlfgrval.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		vtxdlfgrval.a	⊢ 𝐴 = dom 𝐼
4		vtxdlfgrval.d	⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
5		usgrumgr	⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
6	1 2 3 4	vtxdumgrval	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝐷 ‘ 𝑈 ) = ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) )
7	5 6	sylan	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝐷 ‘ 𝑈 ) = ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) )