Theorem iswspthn

Description: An element of the set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018) (Revised by AV, 11-May-2021)

	Ref	Expression
Assertion	iswspthn	$$⊢W\in \left(N\mathrm{WSPathsN}G\right)\leftrightarrow \left(W\in \left(N\mathrm{WWalksN}G\right)\wedge \exists ff\mathrm{SPaths}\left(G\right)W\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq2	$$⊢w=W\to \left(f\mathrm{SPaths}\left(G\right)w\leftrightarrow f\mathrm{SPaths}\left(G\right)W\right)$$
2	1	exbidv	$$⊢w=W\to \left(\exists ff\mathrm{SPaths}\left(G\right)w\leftrightarrow \exists ff\mathrm{SPaths}\left(G\right)W\right)$$
3		wspthsn	$$⊢N\mathrm{WSPathsN}G=\left\{w\in \left(N\mathrm{WWalksN}G\right)|\exists ff\mathrm{SPaths}\left(G\right)w\right\}$$
4	2 3	elrab2	$$⊢W\in \left(N\mathrm{WSPathsN}G\right)\leftrightarrow \left(W\in \left(N\mathrm{WWalksN}G\right)\wedge \exists ff\mathrm{SPaths}\left(G\right)W\right)$$