Metamath Proof Explorer


Theorem uhgrspan1lem1

Description: Lemma 1 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 ) }
Assertion uhgrspan1lem1
|- ( ( V \ { N } ) e. _V /\ ( I |` F ) e. _V )

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 1 fvexi
 |-  V e. _V
5 4 difexi
 |-  ( V \ { N } ) e. _V
6 2 fvexi
 |-  I e. _V
7 6 resex
 |-  ( I |` F ) e. _V
8 5 7 pm3.2i
 |-  ( ( V \ { N } ) e. _V /\ ( I |` F ) e. _V )