Theorem: Matroid.coloop_iff_forall_mem_cl_iff_mem' (Matroid/Loop.lean)

State before:
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
⊢ M.Coloop e ↔ (∀ X ⊆ M.E, e ∈ M.cl X ↔ e ∈ X) ∧ e ∈ M.E
Tactic:
rw [coloop_iff_forall_mem_cl_iff_mem, and_congr_left_iff]
State after:
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
⊢ e ∈ M.E → ((∀ (X : Set α), e ∈ M.cl X ↔ e ∈ X) ↔ ∀ X ⊆ M.E, e ∈ M.cl X ↔ e ∈ X)

State before:
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
⊢ e ∈ M.E → ((∀ (X : Set α), e ∈ M.cl X ↔ e ∈ X) ↔ ∀ X ⊆ M.E, e ∈ M.cl X ↔ e ∈ X)
Tactic:
refine fun he ↦ ⟨fun h X _ ↦ h X, fun h X ↦ ?_⟩
State after:
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X✝ Y Z F C K : Set α
he : e ∈ M.E
h : ∀ X ⊆ M.E, e ∈ M.cl X ↔ e ∈ X
X : Set α
⊢ e ∈ M.cl X ↔ e ∈ X

State before:
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X✝ Y Z F C K : Set α
he : e ∈ M.E
h : ∀ X ⊆ M.E, e ∈ M.cl X ↔ e ∈ X
X : Set α
⊢ e ∈ M.cl X ↔ e ∈ X
Tactic:
rw [← cl_inter_ground, h (X ∩ M.E) inter_subset_right, mem_inter_iff, and_iff_left he]
State after:
no goals