Theorem: Set.pure_seq (GMLInit/Data/Set/Basic.lean)

State before:
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
⊢ (Set.pure f).seq s = Set.map f s
Tactic:
apply Set.ext
State after:
case a
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
⊢ ∀ (x : β), x ∈ (Set.pure f).seq s ↔ x ∈ Set.map f s

State before:
case a
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
⊢ ∀ (x : β), x ∈ (Set.pure f).seq s ↔ x ∈ Set.map f s
Tactic:
intro y
State after:
case a
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
⊢ y ∈ (Set.pure f).seq s ↔ y ∈ Set.map f s

State before:
case a
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
⊢ y ∈ (Set.pure f).seq s ↔ y ∈ Set.map f s
Tactic:
constructor
State after:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
⊢ y ∈ (Set.pure f).seq s → y ∈ Set.map f s

case a.mpr
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
⊢ y ∈ Set.map f s → y ∈ (Set.pure f).seq s

State before:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
⊢ y ∈ (Set.pure f).seq s → y ∈ Set.map f s
Tactic:
intro ⟨_, rfl, h⟩
State after:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
h : y ∈ (fun f => Set.map f s) f
⊢ y ∈ Set.map f s

State before:
case a.mp
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
h : y ∈ (fun f => Set.map f s) f
⊢ y ∈ Set.map f s
Tactic:
exact h
State after:
no goals

State before:
case a.mpr
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
⊢ y ∈ Set.map f s → y ∈ (Set.pure f).seq s
Tactic:
intro ⟨x, hx, h⟩
State after:
case a.mpr
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
x : α
hx : x ∈ s
h : f x = y
⊢ y ∈ (Set.pure f).seq s

State before:
case a.mpr
α : Sort u_1
β : Sort u_2
γ : ?m.9244
f : α → β
s : Set α
y : β
x : α
hx : x ∈ s
h : f x = y
⊢ y ∈ (Set.pure f).seq s
Tactic:
refine ⟨f, rfl, ⟨x, hx, h⟩⟩
State after:
no goals