Metamath Proof Explorer


Theorem clnbupgr

Description: The closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 10-May-2025)

Ref Expression
Hypotheses clnbuhgr.v V = Vtx G
clnbuhgr.e E = Edg G
Assertion clnbupgr G UPGraph N V G ClNeighbVtx N = N n V | N n E

Proof

Step Hyp Ref Expression
1 clnbuhgr.v V = Vtx G
2 clnbuhgr.e E = Edg G
3 1 dfclnbgr4 N V G ClNeighbVtx N = N G NeighbVtx N
4 3 adantl G UPGraph N V G ClNeighbVtx N = N G NeighbVtx N
5 1 2 nbupgr G UPGraph N V G NeighbVtx N = n V N | N n E
6 5 uneq2d G UPGraph N V N G NeighbVtx N = N n V N | N n E
7 rabdif n V | N n E N = n V N | N n E
8 7 eqcomi n V N | N n E = n V | N n E N
9 8 uneq2i N n V N | N n E = N n V | N n E N
10 undif2 N n V | N n E N = N n V | N n E
11 9 10 eqtri N n V N | N n E = N n V | N n E
12 11 a1i G UPGraph N V N n V N | N n E = N n V | N n E
13 4 6 12 3eqtrd G UPGraph N V G ClNeighbVtx N = N n V | N n E