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