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 SubGraph 𝐺 → ( 𝐹 ( Trails ‘ 𝑆 ) 𝑃 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) )

Proof

Step	Hyp	Ref	Expression
1		subgrwlk	⊢ ( 𝑆 SubGraph 𝐺 → ( 𝐹 ( Walks ‘ 𝑆 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) )
2	1	anim1d	⊢ ( 𝑆 SubGraph 𝐺 → ( ( 𝐹 ( Walks ‘ 𝑆 ) 𝑃 ∧ Fun ◡ 𝐹 ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) ) )
3		istrl	⊢ ( 𝐹 ( Trails ‘ 𝑆 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝑆 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
4		istrl	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
5	2 3 4	3imtr4g	⊢ ( 𝑆 SubGraph 𝐺 → ( 𝐹 ( Trails ‘ 𝑆 ) 𝑃 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) )