Metamath Proof Explorer


Theorem struct2grvtx

Description: The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020)

Ref Expression
Hypothesis struct2grvtx.g
|- G = { <. ( Base ` ndx ) , V >. , <. ( .ef ` ndx ) , E >. }
Assertion struct2grvtx
|- ( ( V e. X /\ E e. Y ) -> ( Vtx ` G ) = V )

Proof

Step Hyp Ref Expression
1 struct2grvtx.g
 |-  G = { <. ( Base ` ndx ) , V >. , <. ( .ef ` ndx ) , E >. }
2 edgfndxnn
 |-  ( .ef ` ndx ) e. NN
3 baseltedgf
 |-  ( Base ` ndx ) < ( .ef ` ndx )
4 2 3 1 structvtxval
 |-  ( ( V e. X /\ E e. Y ) -> ( Vtx ` G ) = V )