Theorem wlkRes

Description: Restrictions of walks (i.e. special kinds of walks, for example paths - see pthsfval ) are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017) (Revised by AV, 30-Dec-2020) (Proof shortened by AV, 15-Jan-2021)

	Ref	Expression
Hypothesis	wlkRes.1	⊢ ( 𝑓 ( 𝑊 ‘ 𝐺 ) 𝑝 → 𝑓 ( Walks ‘ 𝐺 ) 𝑝 )
Assertion	wlkRes	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( 𝑊 ‘ 𝐺 ) 𝑝 ∧ 𝜑 ) } ∈ V

Proof

Step	Hyp	Ref	Expression
1		wlkRes.1	⊢ ( 𝑓 ( 𝑊 ‘ 𝐺 ) 𝑝 → 𝑓 ( Walks ‘ 𝐺 ) 𝑝 )
2	1	gen2	⊢ ∀ 𝑓 ∀ 𝑝 ( 𝑓 ( 𝑊 ‘ 𝐺 ) 𝑝 → 𝑓 ( Walks ‘ 𝐺 ) 𝑝 )
3		wksv	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 } ∈ V
4		opabbrex	⊢ ( ( ∀ 𝑓 ∀ 𝑝 ( 𝑓 ( 𝑊 ‘ 𝐺 ) 𝑝 → 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ) ∧ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 } ∈ V ) → { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( 𝑊 ‘ 𝐺 ) 𝑝 ∧ 𝜑 ) } ∈ V )
5	2 3 4	mp2an	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( 𝑊 ‘ 𝐺 ) 𝑝 ∧ 𝜑 ) } ∈ V