Metamath Proof Explorer


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 ) )