Metamath Proof Explorer

Theorem vtxduhgrfiun

Description: The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by Alexander van der Vekens, 21-Jan-2018) (Revised by AV, 7-Dec-2020) (Proof shortened by AV, 19-Feb-2021)

Ref Expression
Hypotheses vtxduhgrun.i ⊢ I = iEdg ⁡ G
vtxduhgrun.j ⊢ J = iEdg ⁡ H
vtxduhgrun.vg ⊢ V = Vtx ⁡ G
vtxduhgrun.vh ⊢ φ → Vtx ⁡ H = V
vtxduhgrun.vu ⊢ φ → Vtx ⁡ U = V
vtxduhgrun.d ⊢ φ → dom ⁡ I ∩ dom ⁡ J = ∅
vtxduhgrun.g ⊢ φ → G ∈ UHGraph
vtxduhgrun.h ⊢ φ → H ∈ UHGraph
vtxduhgrun.n ⊢ φ → N ∈ V
vtxduhgrun.u ⊢ φ → iEdg ⁡ U = I ∪ J
vtxduhgrfiun.a ⊢ φ → dom ⁡ I ∈ Fin
vtxduhgrfiun.b ⊢ φ → dom ⁡ J ∈ Fin
Assertion vtxduhgrfiun ⊢ φ → VtxDeg ⁡ U ⁡ N = VtxDeg ⁡ G ⁡ N + VtxDeg ⁡ H ⁡ N

Proof

Step Hyp Ref Expression
1 vtxduhgrun.i ⊢ I = iEdg ⁡ G
2 vtxduhgrun.j ⊢ J = iEdg ⁡ H
3 vtxduhgrun.vg ⊢ V = Vtx ⁡ G
4 vtxduhgrun.vh ⊢ φ → Vtx ⁡ H = V
5 vtxduhgrun.vu ⊢ φ → Vtx ⁡ U = V
6 vtxduhgrun.d ⊢ φ → dom ⁡ I ∩ dom ⁡ J = ∅
7 vtxduhgrun.g ⊢ φ → G ∈ UHGraph
8 vtxduhgrun.h ⊢ φ → H ∈ UHGraph
9 vtxduhgrun.n ⊢ φ → N ∈ V
10 vtxduhgrun.u ⊢ φ → iEdg ⁡ U = I ∪ J
11 vtxduhgrfiun.a ⊢ φ → dom ⁡ I ∈ Fin
12 vtxduhgrfiun.b ⊢ φ → dom ⁡ J ∈ Fin
13 1 uhgrfun ⊢ G ∈ UHGraph → Fun ⁡ I
14 7 13 syl ⊢ φ → Fun ⁡ I
15 2 uhgrfun ⊢ H ∈ UHGraph → Fun ⁡ J
16 8 15 syl ⊢ φ → Fun ⁡ J
17 1 2 3 4 5 6 14 16 9 10 11 12 vtxdfiun ⊢ φ → VtxDeg ⁡ U ⁡ N = VtxDeg ⁡ G ⁡ N + VtxDeg ⁡ H ⁡ N