Theorem cyclprop

Description: The properties of a cycle: A cycle is a closed path. (Contributed by AV, 31-Jan-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Assertion	cyclprop	$$⊢F\mathrm{Cycles}\left(G\right)P\to \left(F\mathrm{Paths}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		iscycl	$$⊢F\mathrm{Cycles}\left(G\right)P\leftrightarrow \left(F\mathrm{Paths}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)$$
2	1	biimpi	$$⊢F\mathrm{Cycles}\left(G\right)P\to \left(F\mathrm{Paths}\left(G\right)P\wedge P\left(0\right)=P\left(\left|F\right|\right)\right)$$