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	$$⊢fW\left(G\right)p\to f\mathrm{Walks}\left(G\right)p$$
Assertion	wlkResOLD	$$⊢\left\{⟨f,p⟩|\left(fW\left(G\right)p\wedge \phi \right)\right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		wlkResOLD.1	$$⊢fW\left(G\right)p\to f\mathrm{Walks}\left(G\right)p$$
2	1	gen2	$$⊢\forall f\forall p\left(fW\left(G\right)p\to f\mathrm{Walks}\left(G\right)p\right)$$
3		wksv	$$⊢\left\{⟨f,p⟩|f\mathrm{Walks}\left(G\right)p\right\}\in \mathrm{V}$$
4		opabbrex	$$⊢\left(\forall f\forall p\left(fW\left(G\right)p\to f\mathrm{Walks}\left(G\right)p\right)\wedge \left\{⟨f,p⟩|f\mathrm{Walks}\left(G\right)p\right\}\in \mathrm{V}\right)\to \left\{⟨f,p⟩|\left(fW\left(G\right)p\wedge \phi \right)\right\}\in \mathrm{V}$$
5	2 3 4	mp2an	$$⊢\left\{⟨f,p⟩|\left(fW\left(G\right)p\wedge \phi \right)\right\}\in \mathrm{V}$$