Theorem wlkop

Description: A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018) (Revised by AV, 1-Jan-2021)

	Ref	Expression
Assertion	wlkop	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → 𝑊 = ⟨ ( 1st ‘ 𝑊 ) , ( 2nd ‘ 𝑊 ) ⟩ )

Proof

Step	Hyp	Ref	Expression
1		relwlk	⊢ Rel ( Walks ‘ 𝐺 )
2		1st2nd	⊢ ( ( Rel ( Walks ‘ 𝐺 ) ∧ 𝑊 ∈ ( Walks ‘ 𝐺 ) ) → 𝑊 = ⟨ ( 1st ‘ 𝑊 ) , ( 2nd ‘ 𝑊 ) ⟩ )
3	1 2	mpan	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → 𝑊 = ⟨ ( 1st ‘ 𝑊 ) , ( 2nd ‘ 𝑊 ) ⟩ )