ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MonotoneOn.mapsTo_Ici
lemma MonotoneOn.mapsTo_Ici (h : MonotoneOn f (Ici a)) : MapsTo f (Ici a) (Ici (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : MonotoneOn f (Set.Ici a))
Goal: Set.MapsTo f (Set.Ici a) (Set.Ici (f a))
-- MonotoneOn.mapsTo_Iic
lemma MonotoneOn.mapsTo_Iic (h : MonotoneOn f (Iic b)) : MapsTo f (Iic b) (Iic (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : MonotoneOn f (Set.Iic b))
Goal: Set.MapsTo f (Set.Iic b) (Set.Iic (f b))
-- MonotoneOn.mapsTo_Icc
lemma MonotoneOn.mapsTo_Icc (h : MonotoneOn f (Icc a b)) : MapsTo f (Icc a b) (Icc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : MonotoneOn f (Set.Icc a b))
Goal: Set.MapsTo f (Set.Icc a b) (Set.Icc (f a) (f b))
-- AntitoneOn.mapsTo_Ici
lemma AntitoneOn.mapsTo_Ici (h : AntitoneOn f (Ici a)) : MapsTo f (Ici a) (Iic (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : AntitoneOn f (Set.Ici a))
Goal: Set.MapsTo f (Set.Ici a) (Set.Iic (f a))
-- AntitoneOn.mapsTo_Iic
lemma AntitoneOn.mapsTo_Iic (h : AntitoneOn f (Iic b)) : MapsTo f (Iic b) (Ici (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : AntitoneOn f (Set.Iic b))
Goal: Set.MapsTo f (Set.Iic b) (Set.Ici (f b))
-- AntitoneOn.mapsTo_Icc
lemma AntitoneOn.mapsTo_Icc (h : AntitoneOn f (Icc a b)) : MapsTo f (Icc a b) (Icc (f b) (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : AntitoneOn f (Set.Icc a b))
Goal: Set.MapsTo f (Set.Icc a b) (Set.Icc (f b) (f a))
-- StrictMonoOn.mapsTo_Ioi
lemma StrictMonoOn.mapsTo_Ioi (h : StrictMonoOn f (Ici a)) : MapsTo f (Ioi a) (Ioi (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : StrictMonoOn f (Set.Ici a))
Goal: Set.MapsTo f (Set.Ioi a) (Set.Ioi (f a))
-- StrictMonoOn.mapsTo_Iio
lemma StrictMonoOn.mapsTo_Iio (h : StrictMonoOn f (Iic b)) : MapsTo f (Iio b) (Iio (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : StrictMonoOn f (Set.Iic b))
Goal: Set.MapsTo f (Set.Iio b) (Set.Iio (f b))
-- StrictMonoOn.mapsTo_Ioo
lemma StrictMonoOn.mapsTo_Ioo (h : StrictMonoOn f (Icc a b)) :
    MapsTo f (Ioo a b) (Ioo (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : StrictMonoOn f (Set.Icc a b))
Goal: Set.MapsTo f (Set.Ioo a b) (Set.Ioo (f a) (f b))
-- StrictAntiOn.mapsTo_Ioi
lemma StrictAntiOn.mapsTo_Ioi (h : StrictAntiOn f (Ici a)) : MapsTo f (Ioi a) (Iio (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : StrictAntiOn f (Set.Ici a))
Goal: Set.MapsTo f (Set.Ioi a) (Set.Iio (f a))
-- StrictAntiOn.mapsTo_Iio
lemma StrictAntiOn.mapsTo_Iio (h : StrictAntiOn f (Iic b)) : MapsTo f (Iio b) (Ioi (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : StrictAntiOn f (Set.Iic b))
Goal: Set.MapsTo f (Set.Iio b) (Set.Ioi (f b))
-- StrictAntiOn.mapsTo_Ioo
lemma StrictAntiOn.mapsTo_Ioo (h : StrictAntiOn f (Icc a b)) :
    MapsTo f (Ioo a b) (Ioo (f b) (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : StrictAntiOn f (Set.Icc a b))
Goal: Set.MapsTo f (Set.Ioo a b) (Set.Ioo (f b) (f a))
-- Monotone.mapsTo_Ici
lemma Monotone.mapsTo_Ici (h : Monotone f) : MapsTo f (Ici a) (Ici (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : Monotone f)
Goal: Set.MapsTo f (Set.Ici a) (Set.Ici (f a))
-- Monotone.mapsTo_Iic
lemma Monotone.mapsTo_Iic (h : Monotone f) : MapsTo f (Iic b) (Iic (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : Monotone f)
Goal: Set.MapsTo f (Set.Iic b) (Set.Iic (f b))
-- Monotone.mapsTo_Icc
lemma Monotone.mapsTo_Icc (h : Monotone f) : MapsTo f (Icc a b) (Icc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : Monotone f)
Goal: Set.MapsTo f (Set.Icc a b) (Set.Icc (f a) (f b))
-- Antitone.mapsTo_Ici
lemma Antitone.mapsTo_Ici (h : Antitone f) : MapsTo f (Ici a) (Iic (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : Antitone f)
Goal: Set.MapsTo f (Set.Ici a) (Set.Iic (f a))
-- Antitone.mapsTo_Iic
lemma Antitone.mapsTo_Iic (h : Antitone f) : MapsTo f (Iic b) (Ici (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : Antitone f)
Goal: Set.MapsTo f (Set.Iic b) (Set.Ici (f b))
-- Antitone.mapsTo_Icc
lemma Antitone.mapsTo_Icc (h : Antitone f) : MapsTo f (Icc a b) (Icc (f b) (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : Antitone f)
Goal: Set.MapsTo f (Set.Icc a b) (Set.Icc (f b) (f a))
-- StrictMono.mapsTo_Ioi
lemma StrictMono.mapsTo_Ioi (h : StrictMono f) : MapsTo f (Ioi a) (Ioi (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : StrictMono f)
Goal: Set.MapsTo f (Set.Ioi a) (Set.Ioi (f a))
-- StrictMono.mapsTo_Iio
lemma StrictMono.mapsTo_Iio (h : StrictMono f) : MapsTo f (Iio b) (Iio (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : StrictMono f)
Goal: Set.MapsTo f (Set.Iio b) (Set.Iio (f b))
-- StrictMono.mapsTo_Ioo
lemma StrictMono.mapsTo_Ioo (h : StrictMono f) : MapsTo f (Ioo a b) (Ioo (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : StrictMono f)
Goal: Set.MapsTo f (Set.Ioo a b) (Set.Ioo (f a) (f b))
-- StrictAnti.mapsTo_Ioi
lemma StrictAnti.mapsTo_Ioi (h : StrictAnti f) : MapsTo f (Ioi a) (Iio (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : StrictAnti f)
Goal: Set.MapsTo f (Set.Ioi a) (Set.Iio (f a))
-- StrictAnti.mapsTo_Iio
lemma StrictAnti.mapsTo_Iio (h : StrictAnti f) : MapsTo f (Iio b) (Ioi (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : StrictAnti f)
Goal: Set.MapsTo f (Set.Iio b) (Set.Ioi (f b))
-- StrictAnti.mapsTo_Ioo
lemma StrictAnti.mapsTo_Ioo (h : StrictAnti f) : MapsTo f (Ioo a b) (Ioo (f b) (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : StrictAnti f)
Goal: Set.MapsTo f (Set.Ioo a b) (Set.Ioo (f b) (f a))
-- MonotoneOn.image_Ici_subset
lemma MonotoneOn.image_Ici_subset (h : MonotoneOn f (Ici a)) : f '' Ici a ⊆ Ici (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : MonotoneOn f (Set.Ici a))
Goal: f '' Set.Ici a ⊆ Set.Ici (f a)
-- MonotoneOn.image_Iic_subset
lemma MonotoneOn.image_Iic_subset (h : MonotoneOn f (Iic b)) : f '' Iic b ⊆ Iic (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : MonotoneOn f (Set.Iic b))
Goal: f '' Set.Iic b ⊆ Set.Iic (f b)
-- MonotoneOn.image_Icc_subset
lemma MonotoneOn.image_Icc_subset (h : MonotoneOn f (Icc a b)) : f '' Icc a b ⊆ Icc (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : MonotoneOn f (Set.Icc a b))
Goal: f '' Set.Icc a b ⊆ Set.Icc (f a) (f b)
-- AntitoneOn.image_Ici_subset
lemma AntitoneOn.image_Ici_subset (h : AntitoneOn f (Ici a)) : f '' Ici a ⊆ Iic (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : AntitoneOn f (Set.Ici a))
Goal: f '' Set.Ici a ⊆ Set.Iic (f a)
-- AntitoneOn.image_Iic_subset
lemma AntitoneOn.image_Iic_subset (h : AntitoneOn f (Iic b)) : f '' Iic b ⊆ Ici (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : AntitoneOn f (Set.Iic b))
Goal: f '' Set.Iic b ⊆ Set.Ici (f b)
-- AntitoneOn.image_Icc_subset
lemma AntitoneOn.image_Icc_subset (h : AntitoneOn f (Icc a b)) : f '' Icc a b ⊆ Icc (f b) (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : AntitoneOn f (Set.Icc a b))
Goal: f '' Set.Icc a b ⊆ Set.Icc (f b) (f a)
-- StrictMonoOn.image_Ioi_subset
lemma StrictMonoOn.image_Ioi_subset (h : StrictMonoOn f (Ici a)) : f '' Ioi a ⊆ Ioi (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : StrictMonoOn f (Set.Ici a))
Goal: f '' Set.Ioi a ⊆ Set.Ioi (f a)
-- StrictMonoOn.image_Iio_subset
lemma StrictMonoOn.image_Iio_subset (h : StrictMonoOn f (Iic b)) : f '' Iio b ⊆ Iio (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : StrictMonoOn f (Set.Iic b))
Goal: f '' Set.Iio b ⊆ Set.Iio (f b)
-- StrictMonoOn.image_Ioo_subset
lemma StrictMonoOn.image_Ioo_subset (h : StrictMonoOn f (Icc a b)) :
    f '' Ioo a b ⊆ Ioo (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : StrictMonoOn f (Set.Icc a b))
Goal: f '' Set.Ioo a b ⊆ Set.Ioo (f a) (f b)
-- StrictAntiOn.image_Ioi_subset
lemma StrictAntiOn.image_Ioi_subset (h : StrictAntiOn f (Ici a)) : f '' Ioi a ⊆ Iio (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : StrictAntiOn f (Set.Ici a))
Goal: f '' Set.Ioi a ⊆ Set.Iio (f a)
-- StrictAntiOn.image_Iio_subset
lemma StrictAntiOn.image_Iio_subset (h : StrictAntiOn f (Iic b)) : f '' Iio b ⊆ Ioi (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : StrictAntiOn f (Set.Iic b))
Goal: f '' Set.Iio b ⊆ Set.Ioi (f b)
-- StrictAntiOn.image_Ioo_subset
lemma StrictAntiOn.image_Ioo_subset (h : StrictAntiOn f (Icc a b)) :
    f '' Ioo a b ⊆ Ioo (f b) (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : StrictAntiOn f (Set.Icc a b))
Goal: f '' Set.Ioo a b ⊆ Set.Ioo (f b) (f a)
-- Monotone.image_Ici_subset
lemma Monotone.image_Ici_subset (h : Monotone f) : f '' Ici a ⊆ Ici (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : Monotone f)
Goal: f '' Set.Ici a ⊆ Set.Ici (f a)
-- Monotone.image_Iic_subset
lemma Monotone.image_Iic_subset (h : Monotone f) : f '' Iic b ⊆ Iic (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : Monotone f)
Goal: f '' Set.Iic b ⊆ Set.Iic (f b)
-- Monotone.image_Icc_subset
lemma Monotone.image_Icc_subset (h : Monotone f) : f '' Icc a b ⊆ Icc (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : Monotone f)
Goal: f '' Set.Icc a b ⊆ Set.Icc (f a) (f b)
-- Antitone.image_Ici_subset
lemma Antitone.image_Ici_subset (h : Antitone f) : f '' Ici a ⊆ Iic (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : Antitone f)
Goal: f '' Set.Ici a ⊆ Set.Iic (f a)
-- Antitone.image_Iic_subset
lemma Antitone.image_Iic_subset (h : Antitone f) : f '' Iic b ⊆ Ici (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : Antitone f)
Goal: f '' Set.Iic b ⊆ Set.Ici (f b)
-- Antitone.image_Icc_subset
lemma Antitone.image_Icc_subset (h : Antitone f) : f '' Icc a b ⊆ Icc (f b) (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : Antitone f)
Goal: f '' Set.Icc a b ⊆ Set.Icc (f b) (f a)
-- StrictMono.image_Ioi_subset
lemma StrictMono.image_Ioi_subset (h : StrictMono f) : f '' Ioi a ⊆ Ioi (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : StrictMono f)
Goal: f '' Set.Ioi a ⊆ Set.Ioi (f a)
-- StrictMono.image_Iio_subset
lemma StrictMono.image_Iio_subset (h : StrictMono f) : f '' Iio b ⊆ Iio (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : StrictMono f)
Goal: f '' Set.Iio b ⊆ Set.Iio (f b)
-- StrictMono.image_Ioo_subset
lemma StrictMono.image_Ioo_subset (h : StrictMono f) : f '' Ioo a b ⊆ Ioo (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : StrictMono f)
Goal: f '' Set.Ioo a b ⊆ Set.Ioo (f a) (f b)
-- StrictAnti.image_Ioi_subset
lemma StrictAnti.image_Ioi_subset (h : StrictAnti f) : f '' Ioi a ⊆ Iio (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a : α} (h : StrictAnti f)
Goal: f '' Set.Ioi a ⊆ Set.Iio (f a)
-- StrictAnti.image_Iio_subset
lemma StrictAnti.image_Iio_subset (h : StrictAnti f) : f '' Iio b ⊆ Ioi (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {b : α} (h : StrictAnti f)
Goal: f '' Set.Iio b ⊆ Set.Ioi (f b)
-- StrictAnti.image_Ioo_subset
lemma StrictAnti.image_Ioo_subset (h : StrictAnti f) : f '' Ioo a b ⊆ Ioo (f b) (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] {a b : α} (h : StrictAnti f)
Goal: f '' Set.Ioo a b ⊆ Set.Ioo (f b) (f a)
-- StrictMonoOn.mapsTo_Ico
lemma StrictMonoOn.mapsTo_Ico (h : StrictMonoOn f (Icc a b)) :
    MapsTo f (Ico a b) (Ico (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictMonoOn f (Set.Icc a b))
Goal: Set.MapsTo f (Set.Ico a b) (Set.Ico (f a) (f b))
-- StrictMonoOn.mapsTo_Ioc
lemma StrictMonoOn.mapsTo_Ioc (h : StrictMonoOn f (Icc a b)) :
    MapsTo f (Ioc a b) (Ioc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictMonoOn f (Set.Icc a b))
Goal: Set.MapsTo f (Set.Ioc a b) (Set.Ioc (f a) (f b))
-- StrictAntiOn.mapsTo_Ico
lemma StrictAntiOn.mapsTo_Ico (h : StrictAntiOn f (Icc a b)) :
    MapsTo f (Ico a b) (Ioc (f b) (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictAntiOn f (Set.Icc a b))
Goal: Set.MapsTo f (Set.Ico a b) (Set.Ioc (f b) (f a))
-- StrictAntiOn.mapsTo_Ioc
lemma StrictAntiOn.mapsTo_Ioc (h : StrictAntiOn f (Icc a b)) :
    MapsTo f (Ioc a b) (Ico (f b) (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictAntiOn f (Set.Icc a b))
Goal: Set.MapsTo f (Set.Ioc a b) (Set.Ico (f b) (f a))
-- StrictMono.mapsTo_Ico
lemma StrictMono.mapsTo_Ico (h : StrictMono f) : MapsTo f (Ico a b) (Ico (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictMono f)
Goal: Set.MapsTo f (Set.Ico a b) (Set.Ico (f a) (f b))
-- StrictMono.mapsTo_Ioc
lemma StrictMono.mapsTo_Ioc (h : StrictMono f) : MapsTo f (Ioc a b) (Ioc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictMono f)
Goal: Set.MapsTo f (Set.Ioc a b) (Set.Ioc (f a) (f b))
-- StrictAnti.mapsTo_Ico
lemma StrictAnti.mapsTo_Ico (h : StrictAnti f) : MapsTo f (Ico a b) (Ioc (f b) (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictAnti f)
Goal: Set.MapsTo f (Set.Ico a b) (Set.Ioc (f b) (f a))
-- StrictAnti.mapsTo_Ioc
lemma StrictAnti.mapsTo_Ioc (h : StrictAnti f) : MapsTo f (Ioc a b) (Ico (f b) (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictAnti f)
Goal: Set.MapsTo f (Set.Ioc a b) (Set.Ico (f b) (f a))
-- StrictMonoOn.image_Ico_subset
lemma StrictMonoOn.image_Ico_subset (h : StrictMonoOn f (Icc a b)) :
    f '' Ico a b ⊆ Ico (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictMonoOn f (Set.Icc a b))
Goal: f '' Set.Ico a b ⊆ Set.Ico (f a) (f b)
-- StrictMonoOn.image_Ioc_subset
lemma StrictMonoOn.image_Ioc_subset (h : StrictMonoOn f (Icc a b)) :
    f '' Ioc a b ⊆ Ioc (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictMonoOn f (Set.Icc a b))
Goal: f '' Set.Ioc a b ⊆ Set.Ioc (f a) (f b)
-- StrictAntiOn.image_Ico_subset
lemma StrictAntiOn.image_Ico_subset (h : StrictAntiOn f (Icc a b)) :
    f '' Ico a b ⊆ Ioc (f b) (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictAntiOn f (Set.Icc a b))
Goal: f '' Set.Ico a b ⊆ Set.Ioc (f b) (f a)
-- StrictAntiOn.image_Ioc_subset
lemma StrictAntiOn.image_Ioc_subset (h : StrictAntiOn f (Icc a b)) :
    f '' Ioc a b ⊆ Ico (f b) (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictAntiOn f (Set.Icc a b))
Goal: f '' Set.Ioc a b ⊆ Set.Ico (f b) (f a)
-- StrictMono.image_Ico_subset
lemma StrictMono.image_Ico_subset (h : StrictMono f) : f '' Ico a b ⊆ Ico (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictMono f)
Goal: f '' Set.Ico a b ⊆ Set.Ico (f a) (f b)
-- StrictMono.image_Ioc_subset
lemma StrictMono.image_Ioc_subset (h : StrictMono f) : f '' Ioc a b ⊆ Ioc (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictMono f)
Goal: f '' Set.Ioc a b ⊆ Set.Ioc (f a) (f b)
-- StrictAnti.image_Ico_subset
lemma StrictAnti.image_Ico_subset (h : StrictAnti f) : f '' Ico a b ⊆ Ioc (f b) (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictAnti f)
Goal: f '' Set.Ico a b ⊆ Set.Ioc (f b) (f a)
-- StrictAnti.image_Ioc_subset
lemma StrictAnti.image_Ioc_subset (h : StrictAnti f) : f '' Ioc a b ⊆ Ico (f b) (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [PartialOrder α] [Preorder β] {a b : α} (h : StrictAnti f)
Goal: f '' Set.Ioc a b ⊆ Set.Ico (f b) (f a)
-- _private.Mathlib.Order.Interval.Set.Image.0.Set.image_subtype_val_Ixx_Ixi
private lemma image_subtype_val_Ixx_Ixi {p q r : α → α → Prop} {a b : α} (c : {x // p a x ∧ q x b})
    (h : ∀ {x}, r c x → p a x) :
    Subtype.val '' {y : {x // p a x ∧ q x b} | r c.1 y.1} = {y : α | r c.1 y ∧ q y b} 
Arguments: {α : Type u_1} {p q r : α → α → Prop} {a b : α} (c : { x // p a x ∧ q x b }) (h : ∀ {x : α}, r (↑c) x → p a x)
Goal: Subtype.val '' {y | r ↑c ↑y} = {y | r (↑c) y ∧ q y b}
-- _private.Mathlib.Order.Interval.Set.Image.0.Set.image_subtype_val_Ixx_Iix
private lemma image_subtype_val_Ixx_Iix {p q r : α → α → Prop} {a b : α} (c : {x // p a x ∧ q x b})
    (h : ∀ {x}, r x c → q x b) :
    Subtype.val '' {y : {x // p a x ∧ q x b} | r y.1 c.1} = {y : α | p a y ∧ r y c.1} 
Arguments: {α : Type u_1} {p q r : α → α → Prop} {a b : α} (c : { x // p a x ∧ q x b }) (h : ∀ {x : α}, r x ↑c → q x b)
Goal: Subtype.val '' {y | r ↑y ↑c} = {y | p a y ∧ r y ↑c}
-- Set.preimage_subtype_val_Ici
@[simp] lemma preimage_subtype_val_Ici (a : {x // p x}) : (↑) ⁻¹' (Ici a.1) = Ici a 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a : { x // p x })
Goal: Subtype.val ⁻¹' Set.Ici ↑a = Set.Ici a
-- Set.preimage_subtype_val_Iic
@[simp] lemma preimage_subtype_val_Iic (a : {x // p x}) : (↑) ⁻¹' (Iic a.1) = Iic a 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a : { x // p x })
Goal: Subtype.val ⁻¹' Set.Iic ↑a = Set.Iic a
-- Set.preimage_subtype_val_Ioi
@[simp] lemma preimage_subtype_val_Ioi (a : {x // p x}) : (↑) ⁻¹' (Ioi a.1) = Ioi a 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a : { x // p x })
Goal: Subtype.val ⁻¹' Set.Ioi ↑a = Set.Ioi a
-- Set.preimage_subtype_val_Iio
@[simp] lemma preimage_subtype_val_Iio (a : {x // p x}) : (↑) ⁻¹' (Iio a.1) = Iio a 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a : { x // p x })
Goal: Subtype.val ⁻¹' Set.Iio ↑a = Set.Iio a
-- Set.preimage_subtype_val_Icc
@[simp] lemma preimage_subtype_val_Icc (a b : {x // p x}) : (↑) ⁻¹' (Icc a.1 b) = Icc a b 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a b : { x // p x })
Goal: Subtype.val ⁻¹' Set.Icc ↑a ↑b = Set.Icc a b
-- Set.preimage_subtype_val_Ico
@[simp] lemma preimage_subtype_val_Ico (a b : {x // p x}) : (↑) ⁻¹' (Ico a.1 b) = Ico a b 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a b : { x // p x })
Goal: Subtype.val ⁻¹' Set.Ico ↑a ↑b = Set.Ico a b
-- Set.preimage_subtype_val_Ioc
@[simp] lemma preimage_subtype_val_Ioc (a b : {x // p x}) : (↑) ⁻¹' (Ioc a.1 b) = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a b : { x // p x })
Goal: Subtype.val ⁻¹' Set.Ioc ↑a ↑b = Set.Ioc a b
-- Set.preimage_subtype_val_Ioo
@[simp] lemma preimage_subtype_val_Ioo (a b : {x // p x}) : (↑) ⁻¹' (Ioo a.1 b) = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a b : { x // p x })
Goal: Subtype.val ⁻¹' Set.Ioo ↑a ↑b = Set.Ioo a b
-- Set.image_subtype_val_Icc_subset
theorem image_subtype_val_Icc_subset (a b : {x // p x}) :
    Subtype.val '' Icc a b ⊆ Icc a.val b.val 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a b : { x // p x })
Goal: Subtype.val '' Set.Icc a b ⊆ Set.Icc ↑a ↑b
-- Set.image_subtype_val_Ico_subset
theorem image_subtype_val_Ico_subset (a b : {x // p x}) :
    Subtype.val '' Ico a b ⊆ Ico a.val b.val 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a b : { x // p x })
Goal: Subtype.val '' Set.Ico a b ⊆ Set.Ico ↑a ↑b
-- Set.image_subtype_val_Ioc_subset
theorem image_subtype_val_Ioc_subset (a b : {x // p x}) :
    Subtype.val '' Ioc a b ⊆ Ioc a.val b.val 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a b : { x // p x })
Goal: Subtype.val '' Set.Ioc a b ⊆ Set.Ioc ↑a ↑b
-- Set.image_subtype_val_Ioo_subset
theorem image_subtype_val_Ioo_subset (a b : {x // p x}) :
    Subtype.val '' Ioo a b ⊆ Ioo a.val b.val 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a b : { x // p x })
Goal: Subtype.val '' Set.Ioo a b ⊆ Set.Ioo ↑a ↑b
-- Set.image_subtype_val_Iic_subset
theorem image_subtype_val_Iic_subset (a : {x // p x}) :
    Subtype.val '' Iic a ⊆ Iic a.val 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a : { x // p x })
Goal: Subtype.val '' Set.Iic a ⊆ Set.Iic ↑a
-- Set.image_subtype_val_Iio_subset
theorem image_subtype_val_Iio_subset (a : {x // p x}) :
    Subtype.val '' Iio a ⊆ Iio a.val 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a : { x // p x })
Goal: Subtype.val '' Set.Iio a ⊆ Set.Iio ↑a
-- Set.image_subtype_val_Ici_subset
theorem image_subtype_val_Ici_subset (a : {x // p x}) :
    Subtype.val '' Ici a ⊆ Ici a.val 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a : { x // p x })
Goal: Subtype.val '' Set.Ici a ⊆ Set.Ici ↑a
-- Set.image_subtype_val_Ioi_subset
theorem image_subtype_val_Ioi_subset (a : {x // p x}) :
    Subtype.val '' Ioi a ⊆ Ioi a.val 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop} (a : { x // p x })
Goal: Subtype.val '' Set.Ioi a ⊆ Set.Ioi ↑a
-- Set.image_subtype_val_Ici_Iic
@[simp]
lemma image_subtype_val_Ici_Iic {a : α} (b : Ici a) : Subtype.val '' Iic b = Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Ici a))
Goal: Subtype.val '' Set.Iic b = Set.Icc a ↑b
-- Set.image_subtype_val_Ici_Iio
@[simp]
lemma image_subtype_val_Ici_Iio {a : α} (b : Ici a) : Subtype.val '' Iio b = Ico a b 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Ici a))
Goal: Subtype.val '' Set.Iio b = Set.Ico a ↑b
-- Set.image_subtype_val_Ici_Ici
@[simp]
lemma image_subtype_val_Ici_Ici {a : α} (b : Ici a) : Subtype.val '' Ici b = Ici b.1 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Ici a))
Goal: Subtype.val '' Set.Ici b = Set.Ici ↑b
-- Set.image_subtype_val_Ici_Ioi
@[simp]
lemma image_subtype_val_Ici_Ioi {a : α} (b : Ici a) : Subtype.val '' Ioi b = Ioi b.1 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Ici a))
Goal: Subtype.val '' Set.Ioi b = Set.Ioi ↑b
-- Set.image_subtype_val_Iic_Ici
@[simp]
lemma image_subtype_val_Iic_Ici {a : α} (b : Iic a) : Subtype.val '' Ici b = Icc b.1 a 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Iic a))
Goal: Subtype.val '' Set.Ici b = Set.Icc (↑b) a
-- Set.image_subtype_val_Iic_Ioi
@[simp]
lemma image_subtype_val_Iic_Ioi {a : α} (b : Iic a) : Subtype.val '' Ioi b = Ioc b.1 a 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Iic a))
Goal: Subtype.val '' Set.Ioi b = Set.Ioc (↑b) a
-- Set.image_subtype_val_Iic_Iic
@[simp]
lemma image_subtype_val_Iic_Iic {a : α} (b : Iic a) : Subtype.val '' Iic b = Iic b.1 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Iic a))
Goal: Subtype.val '' Set.Iic b = Set.Iic ↑b