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Metamath Proof Explorer

Theorem clnbusgrfi

Description: The closed neighborhood of a vertex in a simple graph with a finite number of edges is a finite set. (Contributed by AV, 10-May-2025)

Ref Expression
Hypotheses clnbusgrf1o.v 
|- V = ( Vtx ` G )
clnbusgrf1o.e 
|- E = ( Edg ` G )
Assertion clnbusgrfi 
|- ( ( G e. USGraph /\ E e. Fin /\ U e. V ) -> ( G ClNeighbVtx U ) e. Fin )

Proof

Step Hyp Ref Expression
1 clnbusgrf1o.v 
 |-  V = ( Vtx ` G )
2 clnbusgrf1o.e 
 |-  E = ( Edg ` G )
3 rabfi 
 |-  ( E e. Fin -> { e e. E | U e. e } e. Fin )
4 3 3ad2ant2 
 |-  ( ( G e. USGraph /\ E e. Fin /\ U e. V ) -> { e e. E | U e. e } e. Fin )
5 1 2 edgusgrclnbfin 
 |-  ( ( G e. USGraph /\ U e. V ) -> ( ( G ClNeighbVtx U ) e. Fin <-> { e e. E | U e. e } e. Fin ) )
6 5 3adant2 
 |-  ( ( G e. USGraph /\ E e. Fin /\ U e. V ) -> ( ( G ClNeighbVtx U ) e. Fin <-> { e e. E | U e. e } e. Fin ) )
7 4 6 mpbird 
 |-  ( ( G e. USGraph /\ E e. Fin /\ U e. V ) -> ( G ClNeighbVtx U ) e. Fin )