Theorem rusgrprop

Description: The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018) (Revised by AV, 18-Dec-2020)

	Ref	Expression
Assertion	rusgrprop	|- ( G RegUSGraph K -> ( G e. USGraph /\ G RegGraph K ) )

Proof

Step	Hyp	Ref	Expression
1		df-rusgr	|- RegUSGraph = { <. g , k >. | ( g e. USGraph /\ g RegGraph k ) }
2	1	bropaex12	|- ( G RegUSGraph K -> ( G e. _V /\ K e. _V ) )
3		isrusgr	|- ( ( G e. _V /\ K e. _V ) -> ( G RegUSGraph K <-> ( G e. USGraph /\ G RegGraph K ) ) )
4	3	biimpd	|- ( ( G e. _V /\ K e. _V ) -> ( G RegUSGraph K -> ( G e. USGraph /\ G RegGraph K ) ) )
5	2 4	mpcom	|- ( G RegUSGraph K -> ( G e. USGraph /\ G RegGraph K ) )