Theorem funvtxdmge2val

Description: The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020) (Revised by AV, 7-Jun-2021) (Revised by AV, 12-Nov-2021)

	Ref	Expression
Assertion	funvtxdmge2val	|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> ( Vtx ` G ) = ( Base ` G ) )

Proof

Step	Hyp	Ref	Expression
1		vtxval	|- ( Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) )
2		fundmge2nop0	|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> -. G e. ( _V X. _V ) )
3	2	iffalsed	|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) = ( Base ` G ) )
4	1 3	syl5eq	|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> ( Vtx ` G ) = ( Base ` G ) )