neighborSet_eq_of_equiv — Mathlib

English

If there is an equivalence between G.neighborSet v and H.neighborSet v for a vertex v, then the two neighbor sets are equal.

Русский

Если существует эквивальентность между множестами соседей в G и H для вершины v, то они равны.

LaTeX

$$$\forall v : V,\ H : \text{Subgraph}(G),\ (G.neighborSet v) \simeq (H.neighborSet v) \Rightarrow H.neighborSet v = G.neighborSet v$$$

Lean4

theorem neighborSet_eq_of_equiv {v : V} {H : Subgraph G} (h : G.neighborSet v ≃ H.neighborSet v)
    (hfin : (G.neighborSet v).Finite) : H.neighborSet v = G.neighborSet v :=
  by
  lift H.neighborSet v to Finset V using h.set_finite_iff.mp hfin with s hs
  lift G.neighborSet v to Finset V using hfin with t ht
  refine congrArg _ <| Finset.eq_of_subset_of_card_le ?_ (Finset.card_eq_of_equiv h).le
  rw [← Finset.coe_subset, hs, ht]
  exact H.neighborSet_subset _

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