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Metamath Proof Explorer

Theorem basvtxval

Description: The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020) (Revised by AV, 12-Nov-2021)

Ref Expression
Hypotheses basvtxval.s 
|- ( ph -> G Struct X )
basvtxval.d 
|- ( ph -> 2 <_ ( # ` dom G ) )
basvtxval.v 
|- ( ph -> V e. Y )
basvtxval.b 
|- ( ph -> <. ( Base ` ndx ) , V >. e. G )
Assertion basvtxval 
|- ( ph -> ( Vtx ` G ) = V )

Proof

Step Hyp Ref Expression
1 basvtxval.s 
 |-  ( ph -> G Struct X )
2 basvtxval.d 
 |-  ( ph -> 2 <_ ( # ` dom G ) )
3 basvtxval.v 
 |-  ( ph -> V e. Y )
4 basvtxval.b 
 |-  ( ph -> <. ( Base ` ndx ) , V >. e. G )
5 structn0fun 
 |-  ( G Struct X -> Fun ( G \ { (/) } ) )
6 1 5 syl 
 |-  ( ph -> Fun ( G \ { (/) } ) )
7 funvtxdmge2val 
 |-  ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> ( Vtx ` G ) = ( Base ` G ) )
8 6 2 7 syl2anc 
 |-  ( ph -> ( Vtx ` G ) = ( Base ` G ) )
9 1 3 4 opelstrbas 
 |-  ( ph -> V = ( Base ` G ) )
10 8 9 eqtr4d 
 |-  ( ph -> ( Vtx ` G ) = V )