File: Mathlib/Order/Interval/Set/WithBotTop.lean

full_name: WithTop.preimage_coe_top
code: @[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) 
goal: WithTop.some ⁻¹' {⊤} = ∅

full_name: WithTop.range_coe
code: theorem range_coe : range (some : α → WithTop α) = Iio ⊤ 
goal: Set.range WithTop.some = Set.Iio ⊤

full_name: WithTop.preimage_coe_Ioi
code: @[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a 
goal: WithTop.some ⁻¹' Set.Ioi ↑a = Set.Ioi a

full_name: WithTop.preimage_coe_Ici
code: @[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a 
goal: WithTop.some ⁻¹' Set.Ici ↑a = Set.Ici a

full_name: WithTop.preimage_coe_Iio
code: @[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a 
goal: WithTop.some ⁻¹' Set.Iio ↑a = Set.Iio a

full_name: WithTop.preimage_coe_Iic
code: @[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a 
goal: WithTop.some ⁻¹' Set.Iic ↑a = Set.Iic a

full_name: WithTop.preimage_coe_Icc
code: @[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b 
goal: WithTop.some ⁻¹' Set.Icc ↑a ↑b = Set.Icc a b

full_name: WithTop.preimage_coe_Ico
code: @[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b 
goal: WithTop.some ⁻¹' Set.Ico ↑a ↑b = Set.Ico a b

full_name: WithTop.preimage_coe_Ioc
code: @[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b 
goal: WithTop.some ⁻¹' Set.Ioc ↑a ↑b = Set.Ioc a b

full_name: WithTop.preimage_coe_Ioo
code: @[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b 
goal: WithTop.some ⁻¹' Set.Ioo ↑a ↑b = Set.Ioo a b

full_name: WithTop.preimage_coe_Iio_top
code: @[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ 
goal: WithTop.some ⁻¹' Set.Iio ⊤ = Set.univ

full_name: WithTop.preimage_coe_Ico_top
code: @[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a 
goal: WithTop.some ⁻¹' Set.Ico ↑a ⊤ = Set.Ici a

full_name: WithTop.preimage_coe_Ioo_top
code: @[simp]
theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a 
goal: WithTop.some ⁻¹' Set.Ioo ↑a ⊤ = Set.Ioi a

full_name: WithTop.image_coe_Ioi
code: theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ 
goal: WithTop.some '' Set.Ioi a = Set.Ioo ↑a ⊤

full_name: WithTop.image_coe_Ici
code: theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ 
goal: WithTop.some '' Set.Ici a = Set.Ico ↑a ⊤

full_name: WithTop.image_coe_Iio
code: theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) 
goal: WithTop.some '' Set.Iio a = Set.Iio ↑a

full_name: WithTop.image_coe_Iic
code: theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) 
goal: WithTop.some '' Set.Iic a = Set.Iic ↑a

full_name: WithTop.image_coe_Icc
code: theorem image_coe_Icc : (some : α → WithTop α) '' Icc a b = Icc (a : WithTop α) b 
goal: WithTop.some '' Set.Icc a b = Set.Icc ↑a ↑b

full_name: WithTop.image_coe_Ico
code: theorem image_coe_Ico : (some : α → WithTop α) '' Ico a b = Ico (a : WithTop α) b 
goal: WithTop.some '' Set.Ico a b = Set.Ico ↑a ↑b

full_name: WithTop.image_coe_Ioc
code: theorem image_coe_Ioc : (some : α → WithTop α) '' Ioc a b = Ioc (a : WithTop α) b 
goal: WithTop.some '' Set.Ioc a b = Set.Ioc ↑a ↑b

full_name: WithTop.image_coe_Ioo
code: theorem image_coe_Ioo : (some : α → WithTop α) '' Ioo a b = Ioo (a : WithTop α) b 
goal: WithTop.some '' Set.Ioo a b = Set.Ioo ↑a ↑b

full_name: WithBot.preimage_coe_bot
code: @[simp]
theorem preimage_coe_bot : (some : α → WithBot α) ⁻¹' {⊥} = (∅ : Set α) 
goal: WithBot.some ⁻¹' {⊥} = ∅

full_name: WithBot.range_coe
code: theorem range_coe : range (some : α → WithBot α) = Ioi ⊥ 
goal: Set.range WithBot.some = Set.Ioi ⊥

full_name: WithBot.preimage_coe_Ioi
code: @[simp]
theorem preimage_coe_Ioi : (some : α → WithBot α) ⁻¹' Ioi a = Ioi a 
goal: WithBot.some ⁻¹' Set.Ioi ↑a = Set.Ioi a

full_name: WithBot.preimage_coe_Ici
code: @[simp]
theorem preimage_coe_Ici : (some : α → WithBot α) ⁻¹' Ici a = Ici a 
goal: WithBot.some ⁻¹' Set.Ici ↑a = Set.Ici a

full_name: WithBot.preimage_coe_Iio
code: @[simp]
theorem preimage_coe_Iio : (some : α → WithBot α) ⁻¹' Iio a = Iio a 
goal: WithBot.some ⁻¹' Set.Iio ↑a = Set.Iio a

full_name: WithBot.preimage_coe_Iic
code: @[simp]
theorem preimage_coe_Iic : (some : α → WithBot α) ⁻¹' Iic a = Iic a 
goal: WithBot.some ⁻¹' Set.Iic ↑a = Set.Iic a

full_name: WithBot.preimage_coe_Icc
code: @[simp]
theorem preimage_coe_Icc : (some : α → WithBot α) ⁻¹' Icc a b = Icc a b 
goal: WithBot.some ⁻¹' Set.Icc ↑a ↑b = Set.Icc a b

full_name: WithBot.preimage_coe_Ico
code: @[simp]
theorem preimage_coe_Ico : (some : α → WithBot α) ⁻¹' Ico a b = Ico a b 
goal: WithBot.some ⁻¹' Set.Ico ↑a ↑b = Set.Ico a b

full_name: WithBot.preimage_coe_Ioc
code: @[simp]
theorem preimage_coe_Ioc : (some : α → WithBot α) ⁻¹' Ioc a b = Ioc a b 
goal: WithBot.some ⁻¹' Set.Ioc ↑a ↑b = Set.Ioc a b

full_name: WithBot.preimage_coe_Ioo
code: @[simp]
theorem preimage_coe_Ioo : (some : α → WithBot α) ⁻¹' Ioo a b = Ioo a b 
goal: WithBot.some ⁻¹' Set.Ioo ↑a ↑b = Set.Ioo a b

full_name: WithBot.preimage_coe_Ioi_bot
code: @[simp]
theorem preimage_coe_Ioi_bot : (some : α → WithBot α) ⁻¹' Ioi ⊥ = univ 
goal: WithBot.some ⁻¹' Set.Ioi ⊥ = Set.univ

full_name: WithBot.preimage_coe_Ioc_bot
code: @[simp]
theorem preimage_coe_Ioc_bot : (some : α → WithBot α) ⁻¹' Ioc ⊥ a = Iic a 
goal: WithBot.some ⁻¹' Set.Ioc ⊥ ↑a = Set.Iic a

full_name: WithBot.preimage_coe_Ioo_bot
code: @[simp]
theorem preimage_coe_Ioo_bot : (some : α → WithBot α) ⁻¹' Ioo ⊥ a = Iio a 
goal: WithBot.some ⁻¹' Set.Ioo ⊥ ↑a = Set.Iio a

full_name: WithBot.image_coe_Iio
code: theorem image_coe_Iio : (some : α → WithBot α) '' Iio a = Ioo (⊥ : WithBot α) a 
goal: WithBot.some '' Set.Iio a = Set.Ioo ⊥ ↑a

full_name: WithBot.image_coe_Iic
code: theorem image_coe_Iic : (some : α → WithBot α) '' Iic a = Ioc (⊥ : WithBot α) a 
goal: WithBot.some '' Set.Iic a = Set.Ioc ⊥ ↑a

full_name: WithBot.image_coe_Ioi
code: theorem image_coe_Ioi : (some : α → WithBot α) '' Ioi a = Ioi (a : WithBot α) 
goal: WithBot.some '' Set.Ioi a = Set.Ioi ↑a

full_name: WithBot.image_coe_Ici
code: theorem image_coe_Ici : (some : α → WithBot α) '' Ici a = Ici (a : WithBot α) 
goal: WithBot.some '' Set.Ici a = Set.Ici ↑a

full_name: WithBot.image_coe_Icc
code: theorem image_coe_Icc : (some : α → WithBot α) '' Icc a b = Icc (a : WithBot α) b 
goal: WithBot.some '' Set.Icc a b = Set.Icc ↑a ↑b

full_name: WithBot.image_coe_Ioc
code: theorem image_coe_Ioc : (some : α → WithBot α) '' Ioc a b = Ioc (a : WithBot α) b 
goal: WithBot.some '' Set.Ioc a b = Set.Ioc ↑a ↑b

full_name: WithBot.image_coe_Ico
code: theorem image_coe_Ico : (some : α → WithBot α) '' Ico a b = Ico (a : WithBot α) b 
goal: WithBot.some '' Set.Ico a b = Set.Ico ↑a ↑b

full_name: WithBot.image_coe_Ioo
code: theorem image_coe_Ioo : (some : α → WithBot α) '' Ioo a b = Ioo (a : WithBot α) b 
goal: WithBot.some '' Set.Ioo a b = Set.Ioo ↑a ↑b