Theorem upgrres1lem2

Description: Lemma 2 for upgrres1 . (Contributed by AV, 7-Nov-2020)

	Ref	Expression
Hypotheses	upgrres1.v	|- V = ( Vtx ` G )
	upgrres1.e	|- E = ( Edg ` G )
	upgrres1.f	|- F = { e e. E | N e/ e }
	upgrres1.s	|- S = <. ( V \ { N } ) , ( _I |` F ) >.
Assertion	upgrres1lem2	|- ( Vtx ` S ) = ( V \ { N } )

Proof

Step	Hyp	Ref	Expression
1		upgrres1.v	|- V = ( Vtx ` G )
2		upgrres1.e	|- E = ( Edg ` G )
3		upgrres1.f	|- F = { e e. E | N e/ e }
4		upgrres1.s	|- S = <. ( V \ { N } ) , ( _I |` F ) >.
5	4	fveq2i	|- ( Vtx ` S ) = ( Vtx ` <. ( V \ { N } ) , ( _I |` F ) >. )
6	1 2 3	upgrres1lem1	|- ( ( V \ { N } ) e. _V /\ ( _I |` F ) e. _V )
7		opvtxfv	|- ( ( ( V \ { N } ) e. _V /\ ( _I |` F ) e. _V ) -> ( Vtx ` <. ( V \ { N } ) , ( _I |` F ) >. ) = ( V \ { N } ) )
8	6 7	ax-mp	|- ( Vtx ` <. ( V \ { N } ) , ( _I |` F ) >. ) = ( V \ { N } )
9	5 8	eqtri	|- ( Vtx ` S ) = ( V \ { N } )