simp only [LE.le, chainUnion, Monad.join, joinM, Set.bind_def, Set.mem_image, exists_and_right,
  exists_eq_right, id_eq, Set.iUnion_exists]
case h
X : Type u_1
inst✝ : ProperInvolutiveInv X
c : Set (InvFreeSets X)
hcChain : IsChain LE.le c
sup : InvFreeSets X := chainUnion c hcChain
A : InvFreeSets X
hAc : A ∈ c
⊢ A ≤ sup
case h
X : Type u_1
inst✝ : ProperInvolutiveInv X
c : Set (InvFreeSets X)
hcChain : IsChain LE.le c
sup : InvFreeSets X := chainUnion c hcChain
A : InvFreeSets X
hAc : A ∈ c
⊢ Set.Subset (↑A) (⋃ i, ⋃ i_1, ⋃ (_ : i_1 ∈ c ∧ ↑i_1 = i), i)