Theorem wspthsswwlknon

Description: The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length between the two vertices. (Contributed by AV, 15-May-2021)

	Ref	Expression
Assertion	wspthsswwlknon	$$⊢A\left(N\mathrm{WSPathsNOn}G\right)B\subseteq A\left(N\mathrm{WWalksNOn}G\right)B$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(G\right)$$
2	1	wspthnonp	$$⊢w\in \left(A\left(N\mathrm{WSPathsNOn}G\right)B\right)\to \left(\left(N\in {\mathbb{N}}_0\wedge G\in \mathrm{V}\right)\wedge \left(A\in \mathrm{Vtx}\left(G\right)\wedge B\in \mathrm{Vtx}\left(G\right)\right)\wedge \left(w\in \left(A\left(N\mathrm{WWalksNOn}G\right)B\right)\wedge \exists ff\left(A\mathrm{SPathsOn}\left(G\right)B\right)w\right)\right)$$
3		simp3l	$$⊢\left(\left(N\in {\mathbb{N}}_0\wedge G\in \mathrm{V}\right)\wedge \left(A\in \mathrm{Vtx}\left(G\right)\wedge B\in \mathrm{Vtx}\left(G\right)\right)\wedge \left(w\in \left(A\left(N\mathrm{WWalksNOn}G\right)B\right)\wedge \exists ff\left(A\mathrm{SPathsOn}\left(G\right)B\right)w\right)\right)\to w\in \left(A\left(N\mathrm{WWalksNOn}G\right)B\right)$$
4	2 3	syl	$$⊢w\in \left(A\left(N\mathrm{WSPathsNOn}G\right)B\right)\to w\in \left(A\left(N\mathrm{WWalksNOn}G\right)B\right)$$
5	4	ssriv	$$⊢A\left(N\mathrm{WSPathsNOn}G\right)B\subseteq A\left(N\mathrm{WWalksNOn}G\right)B$$