Description: A walk as word corresponds to a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018) (Revised by AV, 10-Apr-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | wlkiswwlks | ⊢ ( 𝐺 ∈ USPGraph → ( ∃ 𝑓 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ↔ 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uspgrupgr | ⊢ ( 𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph ) | |
| 2 | wlkiswwlks1 | ⊢ ( 𝐺 ∈ UPGraph → ( 𝑓 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) ) | |
| 3 | 1 2 | syl | ⊢ ( 𝐺 ∈ USPGraph → ( 𝑓 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) ) |
| 4 | 3 | exlimdv | ⊢ ( 𝐺 ∈ USPGraph → ( ∃ 𝑓 𝑓 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) ) |
| 5 | wlkiswwlks2 | ⊢ ( 𝐺 ∈ USPGraph → ( 𝑃 ∈ ( WWalks ‘ 𝐺 ) → ∃ 𝑓 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ) ) | |
| 6 | 4 5 | impbid | ⊢ ( 𝐺 ∈ USPGraph → ( ∃ 𝑓 𝑓 ( Walks ‘ 𝐺 ) 𝑃 ↔ 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) ) |