Theorem wlkResOLD

Description: Obsolete version of opabresex2 as of 13-Dec-2024. (Contributed by Alexander van der Vekens, 1-Nov-2017) (Revised by AV, 30-Dec-2020) (Proof shortened by AV, 15-Jan-2021) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		wlkResOLD.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