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Metamath Proof Explorer

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 ) } ) )