Theorem: Set.iUnionLift_unary
File: Mathlib/Data/Set/UnionLift.lean

tactic: subst hT'
state_before: α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf :
  ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),
    f i { val := x, property := hxi } = f j { val := x, property := hxj }
T : Set α
hT : T ⊆ iUnion S
hT' : T = iUnion S
u : ↑T → ↑T
ui : (i : ι) → ↑(S i) → ↑(S i)
hui : ∀ (i : ι) (x : ↑(S i)), u (inclusion (_ : S i ⊆ T) x) = inclusion (_ : S i ⊆ T) (ui i x)
uβ : β → β
h : ∀ (i : ι) (x : ↑(S i)), f i (ui i x) = uβ (f i x)
x : ↑T
⊢ iUnionLift S f hf T (_ : T ≤ iUnion S) (u x) = uβ (iUnionLift S f hf T (_ : T ≤ iUnion S) x)

tactic: cases' Set.mem_iUnion.1 x.prop with i hi
state_before: α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf :
  ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),
    f i { val := x, property := hxi } = f j { val := x, property := hxj }
ui : (i : ι) → ↑(S i) → ↑(S i)
uβ : β → β
h : ∀ (i : ι) (x : ↑(S i)), f i (ui i x) = uβ (f i x)
hT : iUnion S ⊆ iUnion S
u : ↑(iUnion S) → ↑(iUnion S)
hui : ∀ (i : ι) (x : ↑(S i)), u (inclusion (_ : S i ⊆ iUnion S) x) = inclusion (_ : S i ⊆ iUnion S) (ui i x)
x : ↑(iUnion S)
⊢ iUnionLift S f hf (iUnion S) (_ : iUnion S ≤ iUnion S) (u x) =
    uβ (iUnionLift S f hf (iUnion S) (_ : iUnion S ≤ iUnion S) x)

tactic: rw [iUnionLift_of_mem x hi, ← h i]
state_before: case intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf :
  ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),
    f i { val := x, property := hxi } = f j { val := x, property := hxj }
ui : (i : ι) → ↑(S i) → ↑(S i)
uβ : β → β
h : ∀ (i : ι) (x : ↑(S i)), f i (ui i x) = uβ (f i x)
hT : iUnion S ⊆ iUnion S
u : ↑(iUnion S) → ↑(iUnion S)
hui : ∀ (i : ι) (x : ↑(S i)), u (inclusion (_ : S i ⊆ iUnion S) x) = inclusion (_ : S i ⊆ iUnion S) (ui i x)
x : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
⊢ iUnionLift S f hf (iUnion S) (_ : iUnion S ≤ iUnion S) (u x) =
    uβ (iUnionLift S f hf (iUnion S) (_ : iUnion S ≤ iUnion S) x)

tactic: have : x = Set.inclusion (Set.subset_iUnion S i) ⟨x, hi⟩ := by
  cases x
  rfl
state_before: case intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf :
  ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),
    f i { val := x, property := hxi } = f j { val := x, property := hxj }
ui : (i : ι) → ↑(S i) → ↑(S i)
uβ : β → β
h : ∀ (i : ι) (x : ↑(S i)), f i (ui i x) = uβ (f i x)
hT : iUnion S ⊆ iUnion S
u : ↑(iUnion S) → ↑(iUnion S)
hui : ∀ (i : ι) (x : ↑(S i)), u (inclusion (_ : S i ⊆ iUnion S) x) = inclusion (_ : S i ⊆ iUnion S) (ui i x)
x : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
⊢ iUnionLift S f hf (iUnion S) (_ : iUnion S ≤ iUnion S) (u x) = f i (ui i { val := ↑x, property := hi })

tactic: conv_lhs => rw [this, hui, iUnionLift_inclusion]
state_before: case intro
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf :
  ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),
    f i { val := x, property := hxi } = f j { val := x, property := hxj }
ui : (i : ι) → ↑(S i) → ↑(S i)
uβ : β → β
h : ∀ (i : ι) (x : ↑(S i)), f i (ui i x) = uβ (f i x)
hT : iUnion S ⊆ iUnion S
u : ↑(iUnion S) → ↑(iUnion S)
hui : ∀ (i : ι) (x : ↑(S i)), u (inclusion (_ : S i ⊆ iUnion S) x) = inclusion (_ : S i ⊆ iUnion S) (ui i x)
x : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
this : x = inclusion (_ : S i ⊆ ⋃ (i : ι), S i) { val := ↑x, property := hi }
⊢ iUnionLift S f hf (iUnion S) (_ : iUnion S ≤ iUnion S) (u x) = f i (ui i { val := ↑x, property := hi })

tactic: cases x
state_before: α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf :
  ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),
    f i { val := x, property := hxi } = f j { val := x, property := hxj }
ui : (i : ι) → ↑(S i) → ↑(S i)
uβ : β → β
h : ∀ (i : ι) (x : ↑(S i)), f i (ui i x) = uβ (f i x)
hT : iUnion S ⊆ iUnion S
u : ↑(iUnion S) → ↑(iUnion S)
hui : ∀ (i : ι) (x : ↑(S i)), u (inclusion (_ : S i ⊆ iUnion S) x) = inclusion (_ : S i ⊆ iUnion S) (ui i x)
x : ↑(iUnion S)
i : ι
hi : ↑x ∈ S i
⊢ x = inclusion (_ : S i ⊆ ⋃ (i : ι), S i) { val := ↑x, property := hi }

tactic: rfl
state_before: case mk
α : Type u_1
ι : Sort u_2
β : Sort u_3
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf :
  ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),
    f i { val := x, property := hxi } = f j { val := x, property := hxj }
ui : (i : ι) → ↑(S i) → ↑(S i)
uβ : β → β
h : ∀ (i : ι) (x : ↑(S i)), f i (ui i x) = uβ (f i x)
hT : iUnion S ⊆ iUnion S
u : ↑(iUnion S) → ↑(iUnion S)
hui : ∀ (i : ι) (x : ↑(S i)), u (inclusion (_ : S i ⊆ iUnion S) x) = inclusion (_ : S i ⊆ iUnion S) (ui i x)
i : ι
val✝ : α
property✝ : val✝ ∈ iUnion S
hi : ↑{ val := val✝, property := property✝ } ∈ S i
⊢ { val := val✝, property := property✝ } =
    inclusion (_ : S i ⊆ ⋃ (i : ι), S i) { val := ↑{ val := val✝, property := property✝ }, property := hi }