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Metamath Proof Explorer

Theorem opfusgr

Description: A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020)

Ref Expression
Assertion opfusgr 
|- ( ( V e. X /\ E e. Y ) -> ( <. V , E >. e. FinUSGraph <-> ( <. V , E >. e. USGraph /\ V e. Fin ) ) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Vtx ` <. V , E >. ) = ( Vtx ` <. V , E >. )
2 1 isfusgr 
 |-  ( <. V , E >. e. FinUSGraph <-> ( <. V , E >. e. USGraph /\ ( Vtx ` <. V , E >. ) e. Fin ) )
3 opvtxfv 
 |-  ( ( V e. X /\ E e. Y ) -> ( Vtx ` <. V , E >. ) = V )
4 3 eleq1d 
 |-  ( ( V e. X /\ E e. Y ) -> ( ( Vtx ` <. V , E >. ) e. Fin <-> V e. Fin ) )
5 4 anbi2d 
 |-  ( ( V e. X /\ E e. Y ) -> ( ( <. V , E >. e. USGraph /\ ( Vtx ` <. V , E >. ) e. Fin ) <-> ( <. V , E >. e. USGraph /\ V e. Fin ) ) )
6 2 5 bitrid 
 |-  ( ( V e. X /\ E e. Y ) -> ( <. V , E >. e. FinUSGraph <-> ( <. V , E >. e. USGraph /\ V e. Fin ) ) )