Theorem spthsfval

Description: The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017) (Revised by AV, 9-Jan-2021) (Revised by AV, 29-Oct-2021)

	Ref	Expression
Assertion	spthsfval	⊢ ( SPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑝 ) }

Proof

Step	Hyp	Ref	Expression
1		biidd	⊢ ( ( ⊤ ∧ 𝑔 = 𝐺 ) → ( Fun ◡ 𝑝 ↔ Fun ◡ 𝑝 ) )
2		wksv	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 } ∈ V
3		trliswlk	⊢ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 → 𝑓 ( Walks ‘ 𝐺 ) 𝑝 )
4	3	ssopab2i	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ⊆ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 }
5	2 4	ssexi	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ∈ V
6	5	a1i	⊢ ( ⊤ → { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ∈ V )
7		df-spths	⊢ SPaths = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ Fun ◡ 𝑝 ) } )
8	1 6 7	fvmptopab	⊢ ( ⊤ → ( SPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑝 ) } )
9	8	mptru	⊢ ( SPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑝 ) }