Theorem vtxdumgrval

Description: The value of the vertex degree function for a multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017) (Revised by AV, 23-Feb-2021)

	Ref	Expression
Hypotheses	vtxdlfgrval.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	vtxdlfgrval.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	vtxdlfgrval.a	⊢ 𝐴 = dom 𝐼
	vtxdlfgrval.d	⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
Assertion	vtxdumgrval	⊢ ( ( 𝐺 ∈ UMGraph ∧ 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	1 2	umgrislfupgr	⊢ ( 𝐺 ∈ UMGraph ↔ ( 𝐺 ∈ UPGraph ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) )
6	3	eqcomi	⊢ dom 𝐼 = 𝐴
7	6	feq2i	⊢ ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ↔ 𝐼 : 𝐴 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } )
8	7	biimpi	⊢ ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → 𝐼 : 𝐴 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } )
9	5 8	simplbiim	⊢ ( 𝐺 ∈ UMGraph → 𝐼 : 𝐴 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } )
10	1 2 3 4	vtxdlfgrval	⊢ ( ( 𝐼 : 𝐴 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ∧ 𝑈 ∈ 𝑉 ) → ( 𝐷 ‘ 𝑈 ) = ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) )
11	9 10	sylan	⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝐷 ‘ 𝑈 ) = ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) )