Theorem 0wlkonlem2

Description: Lemma 2 for 0wlkon and 0trlon . (Contributed by AV, 3-Jan-2021) (Revised by AV, 23-Mar-2021)

	Ref	Expression
Hypothesis	0wlk.v	|- V = ( Vtx ` G )
Assertion	0wlkonlem2	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> P e. ( V ^pm ( 0 ... 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		0wlk.v	|- V = ( Vtx ` G )
2		ovex	|- ( 0 ... 0 ) e. _V
3	1	fvexi	|- V e. _V
4		simpl	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> P : ( 0 ... 0 ) --> V )
5		fpmg	|- ( ( ( 0 ... 0 ) e. _V /\ V e. _V /\ P : ( 0 ... 0 ) --> V ) -> P e. ( V ^pm ( 0 ... 0 ) ) )
6	2 3 4 5	mp3an12i	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> P e. ( V ^pm ( 0 ... 0 ) ) )