Theorem uhgrspan1lem2

Description: Lemma 2 for uhgrspan1 . (Contributed by AV, 19-Nov-2020)

	Ref	Expression
Hypotheses	uhgrspan1.v	|- V = ( Vtx ` G )
	uhgrspan1.i	|- I = ( iEdg ` G )
	uhgrspan1.f	|- F = { i e. dom I | N e/ ( I ` i ) }
	uhgrspan1.s	|- S = <. ( V \ { N } ) , ( I |` F ) >.
Assertion	uhgrspan1lem2	|- ( Vtx ` S ) = ( V \ { N } )

Proof

Step	Hyp	Ref	Expression
1		uhgrspan1.v	|- V = ( Vtx ` G )
2		uhgrspan1.i	|- I = ( iEdg ` G )
3		uhgrspan1.f	|- F = { i e. dom I | N e/ ( I ` i ) }
4		uhgrspan1.s	|- S = <. ( V \ { N } ) , ( I |` F ) >.
5	4	fveq2i	|- ( Vtx ` S ) = ( Vtx ` <. ( V \ { N } ) , ( I |` F ) >. )
6	1 2 3	uhgrspan1lem1	|- ( ( 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 } )