Description: An induced subgraph of a multigraph is a multigraph. (Contributed by AV, 15-May-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | isubgrupgr.v | |- V = ( Vtx ` G ) |
|
| Assertion | isubgrumgr | |- ( ( G e. UMGraph /\ S C_ V ) -> ( G ISubGr S ) e. UMGraph ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isubgrupgr.v | |- V = ( Vtx ` G ) |
|
| 2 | umgruhgr | |- ( G e. UMGraph -> G e. UHGraph ) |
|
| 3 | 1 | isubgrsubgr | |- ( ( G e. UHGraph /\ S C_ V ) -> ( G ISubGr S ) SubGraph G ) |
| 4 | 2 3 | sylan | |- ( ( G e. UMGraph /\ S C_ V ) -> ( G ISubGr S ) SubGraph G ) |
| 5 | subumgr | |- ( ( G e. UMGraph /\ ( G ISubGr S ) SubGraph G ) -> ( G ISubGr S ) e. UMGraph ) |
|
| 6 | 4 5 | syldan | |- ( ( G e. UMGraph /\ S C_ V ) -> ( G ISubGr S ) e. UMGraph ) |