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