Metamath Proof Explorer


Theorem cplgruvtxb

Description: A graph G is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017) (Revised by AV, 15-Feb-2022)

Ref Expression
Hypothesis cplgruvtxb.v V = Vtx G
Assertion cplgruvtxb G W G ComplGraph UnivVtx G = V

Proof

Step Hyp Ref Expression
1 cplgruvtxb.v V = Vtx G
2 fveq2 g = G UnivVtx g = UnivVtx G
3 fveq2 g = G Vtx g = Vtx G
4 3 1 eqtr4di g = G Vtx g = V
5 2 4 eqeq12d g = G UnivVtx g = Vtx g UnivVtx G = V
6 df-cplgr ComplGraph = g | UnivVtx g = Vtx g
7 5 6 elab2g G W G ComplGraph UnivVtx G = V