Metamath Proof Explorer


Theorem opvtxfv

Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020)

Ref Expression
Assertion opvtxfv
|- ( ( V e. X /\ E e. Y ) -> ( Vtx ` <. V , E >. ) = V )

Proof

Step Hyp Ref Expression
1 opelvvg
 |-  ( ( V e. X /\ E e. Y ) -> <. V , E >. e. ( _V X. _V ) )
2 opvtxval
 |-  ( <. V , E >. e. ( _V X. _V ) -> ( Vtx ` <. V , E >. ) = ( 1st ` <. V , E >. ) )
3 1 2 syl
 |-  ( ( V e. X /\ E e. Y ) -> ( Vtx ` <. V , E >. ) = ( 1st ` <. V , E >. ) )
4 op1stg
 |-  ( ( V e. X /\ E e. Y ) -> ( 1st ` <. V , E >. ) = V )
5 3 4 eqtrd
 |-  ( ( V e. X /\ E e. Y ) -> ( Vtx ` <. V , E >. ) = V )