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Metamath Proof Explorer

Theorem umgr2v2evtx

Description: The set of vertices in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020)

Ref Expression
Hypothesis umgr2v2evtx.g 
|- G = <. V , { <. 0 , { A , B } >. , <. 1 , { A , B } >. } >.
Assertion umgr2v2evtx 
|- ( V e. W -> ( Vtx ` G ) = V )

Proof

Step Hyp Ref Expression
1 umgr2v2evtx.g 
 |-  G = <. V , { <. 0 , { A , B } >. , <. 1 , { A , B } >. } >.
2 1 fveq2i 
 |-  ( Vtx ` G ) = ( Vtx ` <. V , { <. 0 , { A , B } >. , <. 1 , { A , B } >. } >. )
3 prex 
 |-  { <. 0 , { A , B } >. , <. 1 , { A , B } >. } e. _V
4 opvtxfv 
 |-  ( ( V e. W /\ { <. 0 , { A , B } >. , <. 1 , { A , B } >. } e. _V ) -> ( Vtx ` <. V , { <. 0 , { A , B } >. , <. 1 , { A , B } >. } >. ) = V )
5 3 4 mpan2 
 |-  ( V e. W -> ( Vtx ` <. V , { <. 0 , { A , B } >. , <. 1 , { A , B } >. } >. ) = V )
6 2 5 eqtrid 
 |-  ( V e. W -> ( Vtx ` G ) = V )