Theorem subgrtrl

Description: If a trail exists in a subgraph of a graph G , then that trail also exists in G . (Contributed by BTernaryTau, 22-Oct-2023)

	Ref	Expression
Assertion	subgrtrl	$$⊢S\mathrm{SubGraph}G\to \left(F\mathrm{Trails}\left(S\right)P\to F\mathrm{Trails}\left(G\right)P\right)$$

Proof

Step	Hyp	Ref	Expression
1		subgrwlk	$$⊢S\mathrm{SubGraph}G\to \left(F\mathrm{Walks}\left(S\right)P\to F\mathrm{Walks}\left(G\right)P\right)$$
2	1	anim1d	$$⊢S\mathrm{SubGraph}G\to \left(\left(F\mathrm{Walks}\left(S\right)P\wedge \mathrm{Fun}{F}^{-1}\right)\to \left(F\mathrm{Walks}\left(G\right)P\wedge \mathrm{Fun}{F}^{-1}\right)\right)$$
3		istrl	$$⊢F\mathrm{Trails}\left(S\right)P\leftrightarrow \left(F\mathrm{Walks}\left(S\right)P\wedge \mathrm{Fun}{F}^{-1}\right)$$
4		istrl	$$⊢F\mathrm{Trails}\left(G\right)P\leftrightarrow \left(F\mathrm{Walks}\left(G\right)P\wedge \mathrm{Fun}{F}^{-1}\right)$$
5	2 3 4	3imtr4g	$$⊢S\mathrm{SubGraph}G\to \left(F\mathrm{Trails}\left(S\right)P\to F\mathrm{Trails}\left(G\right)P\right)$$