Theorem funvtxdm2val

Description: The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020) (Revised by AV, 7-Jun-2021) (Revised by AV, 12-Nov-2021)

	Ref	Expression
Hypotheses	funvtxdm2val.a	|- A e. _V
	funvtxdm2val.b	|- B e. _V
Assertion	funvtxdm2val	|- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> ( Vtx ` G ) = ( Base ` G ) )

Proof

Step	Hyp	Ref	Expression
1		funvtxdm2val.a	|- A e. _V
2		funvtxdm2val.b	|- B e. _V
3		vtxval	|- ( Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) )
4	1 2	fun2dmnop0	|- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _V ) )
5	4	iffalsed	|- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) = ( Base ` G ) )
6	3 5	syl5eq	|- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> ( Vtx ` G ) = ( Base ` G ) )