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

full_name: MonotoneOn.mapsTo_Ici
code: lemma MonotoneOn.mapsTo_Ici (h : MonotoneOn f (Ici a)) : MapsTo f (Ici a) (Ici (f a)) 
goal: Set.MapsTo f (Set.Ici a) (Set.Ici (f a))

full_name: MonotoneOn.mapsTo_Iic
code: lemma MonotoneOn.mapsTo_Iic (h : MonotoneOn f (Iic b)) : MapsTo f (Iic b) (Iic (f b)) 
goal: Set.MapsTo f (Set.Iic b) (Set.Iic (f b))

full_name: MonotoneOn.mapsTo_Icc
code: lemma MonotoneOn.mapsTo_Icc (h : MonotoneOn f (Icc a b)) : MapsTo f (Icc a b) (Icc (f a) (f b)) 
goal: Set.MapsTo f (Set.Icc a b) (Set.Icc (f a) (f b))

full_name: AntitoneOn.mapsTo_Ici
code: lemma AntitoneOn.mapsTo_Ici (h : AntitoneOn f (Ici a)) : MapsTo f (Ici a) (Iic (f a)) 
goal: Set.MapsTo f (Set.Ici a) (Set.Iic (f a))

full_name: AntitoneOn.mapsTo_Iic
code: lemma AntitoneOn.mapsTo_Iic (h : AntitoneOn f (Iic b)) : MapsTo f (Iic b) (Ici (f b)) 
goal: Set.MapsTo f (Set.Iic b) (Set.Ici (f b))

full_name: AntitoneOn.mapsTo_Icc
code: lemma AntitoneOn.mapsTo_Icc (h : AntitoneOn f (Icc a b)) : MapsTo f (Icc a b) (Icc (f b) (f a)) 
goal: Set.MapsTo f (Set.Icc a b) (Set.Icc (f b) (f a))

full_name: StrictMonoOn.mapsTo_Ioi
code: lemma StrictMonoOn.mapsTo_Ioi (h : StrictMonoOn f (Ici a)) : MapsTo f (Ioi a) (Ioi (f a)) 
goal: Set.MapsTo f (Set.Ioi a) (Set.Ioi (f a))

full_name: StrictMonoOn.mapsTo_Iio
code: lemma StrictMonoOn.mapsTo_Iio (h : StrictMonoOn f (Iic b)) : MapsTo f (Iio b) (Iio (f b)) 
goal: Set.MapsTo f (Set.Iio b) (Set.Iio (f b))

full_name: StrictMonoOn.mapsTo_Ioo
code: lemma StrictMonoOn.mapsTo_Ioo (h : StrictMonoOn f (Icc a b)) :
    MapsTo f (Ioo a b) (Ioo (f a) (f b)) 
goal: Set.MapsTo f (Set.Ioo a b) (Set.Ioo (f a) (f b))

full_name: StrictAntiOn.mapsTo_Ioi
code: lemma StrictAntiOn.mapsTo_Ioi (h : StrictAntiOn f (Ici a)) : MapsTo f (Ioi a) (Iio (f a)) 
goal: Set.MapsTo f (Set.Ioi a) (Set.Iio (f a))

full_name: StrictAntiOn.mapsTo_Iio
code: lemma StrictAntiOn.mapsTo_Iio (h : StrictAntiOn f (Iic b)) : MapsTo f (Iio b) (Ioi (f b)) 
goal: Set.MapsTo f (Set.Iio b) (Set.Ioi (f b))

full_name: StrictAntiOn.mapsTo_Ioo
code: lemma StrictAntiOn.mapsTo_Ioo (h : StrictAntiOn f (Icc a b)) :
    MapsTo f (Ioo a b) (Ioo (f b) (f a)) 
goal: Set.MapsTo f (Set.Ioo a b) (Set.Ioo (f b) (f a))

full_name: Monotone.mapsTo_Ici
code: lemma Monotone.mapsTo_Ici (h : Monotone f) : MapsTo f (Ici a) (Ici (f a)) 
goal: Set.MapsTo f (Set.Ici a) (Set.Ici (f a))

full_name: Monotone.mapsTo_Iic
code: lemma Monotone.mapsTo_Iic (h : Monotone f) : MapsTo f (Iic b) (Iic (f b)) 
goal: Set.MapsTo f (Set.Iic b) (Set.Iic (f b))

full_name: Monotone.mapsTo_Icc
code: lemma Monotone.mapsTo_Icc (h : Monotone f) : MapsTo f (Icc a b) (Icc (f a) (f b)) 
goal: Set.MapsTo f (Set.Icc a b) (Set.Icc (f a) (f b))

full_name: Antitone.mapsTo_Ici
code: lemma Antitone.mapsTo_Ici (h : Antitone f) : MapsTo f (Ici a) (Iic (f a)) 
goal: Set.MapsTo f (Set.Ici a) (Set.Iic (f a))

full_name: Antitone.mapsTo_Iic
code: lemma Antitone.mapsTo_Iic (h : Antitone f) : MapsTo f (Iic b) (Ici (f b)) 
goal: Set.MapsTo f (Set.Iic b) (Set.Ici (f b))

full_name: Antitone.mapsTo_Icc
code: lemma Antitone.mapsTo_Icc (h : Antitone f) : MapsTo f (Icc a b) (Icc (f b) (f a)) 
goal: Set.MapsTo f (Set.Icc a b) (Set.Icc (f b) (f a))

full_name: StrictMono.mapsTo_Ioi
code: lemma StrictMono.mapsTo_Ioi (h : StrictMono f) : MapsTo f (Ioi a) (Ioi (f a)) 
goal: Set.MapsTo f (Set.Ioi a) (Set.Ioi (f a))

full_name: StrictMono.mapsTo_Iio
code: lemma StrictMono.mapsTo_Iio (h : StrictMono f) : MapsTo f (Iio b) (Iio (f b)) 
goal: Set.MapsTo f (Set.Iio b) (Set.Iio (f b))

full_name: StrictMono.mapsTo_Ioo
code: lemma StrictMono.mapsTo_Ioo (h : StrictMono f) : MapsTo f (Ioo a b) (Ioo (f a) (f b)) 
goal: Set.MapsTo f (Set.Ioo a b) (Set.Ioo (f a) (f b))

full_name: StrictAnti.mapsTo_Ioi
code: lemma StrictAnti.mapsTo_Ioi (h : StrictAnti f) : MapsTo f (Ioi a) (Iio (f a)) 
goal: Set.MapsTo f (Set.Ioi a) (Set.Iio (f a))

full_name: StrictAnti.mapsTo_Iio
code: lemma StrictAnti.mapsTo_Iio (h : StrictAnti f) : MapsTo f (Iio b) (Ioi (f b)) 
goal: Set.MapsTo f (Set.Iio b) (Set.Ioi (f b))

full_name: StrictAnti.mapsTo_Ioo
code: lemma StrictAnti.mapsTo_Ioo (h : StrictAnti f) : MapsTo f (Ioo a b) (Ioo (f b) (f a)) 
goal: Set.MapsTo f (Set.Ioo a b) (Set.Ioo (f b) (f a))

full_name: MonotoneOn.image_Ici_subset
code: lemma MonotoneOn.image_Ici_subset (h : MonotoneOn f (Ici a)) : f '' Ici a ⊆ Ici (f a) 
goal: f '' Set.Ici a ⊆ Set.Ici (f a)

full_name: MonotoneOn.image_Iic_subset
code: lemma MonotoneOn.image_Iic_subset (h : MonotoneOn f (Iic b)) : f '' Iic b ⊆ Iic (f b) 
goal: f '' Set.Iic b ⊆ Set.Iic (f b)

full_name: MonotoneOn.image_Icc_subset
code: lemma MonotoneOn.image_Icc_subset (h : MonotoneOn f (Icc a b)) : f '' Icc a b ⊆ Icc (f a) (f b) 
goal: f '' Set.Icc a b ⊆ Set.Icc (f a) (f b)

full_name: AntitoneOn.image_Ici_subset
code: lemma AntitoneOn.image_Ici_subset (h : AntitoneOn f (Ici a)) : f '' Ici a ⊆ Iic (f a) 
goal: f '' Set.Ici a ⊆ Set.Iic (f a)

full_name: AntitoneOn.image_Iic_subset
code: lemma AntitoneOn.image_Iic_subset (h : AntitoneOn f (Iic b)) : f '' Iic b ⊆ Ici (f b) 
goal: f '' Set.Iic b ⊆ Set.Ici (f b)

full_name: AntitoneOn.image_Icc_subset
code: lemma AntitoneOn.image_Icc_subset (h : AntitoneOn f (Icc a b)) : f '' Icc a b ⊆ Icc (f b) (f a) 
goal: f '' Set.Icc a b ⊆ Set.Icc (f b) (f a)

full_name: StrictMonoOn.image_Ioi_subset
code: lemma StrictMonoOn.image_Ioi_subset (h : StrictMonoOn f (Ici a)) : f '' Ioi a ⊆ Ioi (f a) 
goal: f '' Set.Ioi a ⊆ Set.Ioi (f a)

full_name: StrictMonoOn.image_Iio_subset
code: lemma StrictMonoOn.image_Iio_subset (h : StrictMonoOn f (Iic b)) : f '' Iio b ⊆ Iio (f b) 
goal: f '' Set.Iio b ⊆ Set.Iio (f b)

full_name: StrictMonoOn.image_Ioo_subset
code: lemma StrictMonoOn.image_Ioo_subset (h : StrictMonoOn f (Icc a b)) :
    f '' Ioo a b ⊆ Ioo (f a) (f b) 
goal: f '' Set.Ioo a b ⊆ Set.Ioo (f a) (f b)

full_name: StrictAntiOn.image_Ioi_subset
code: lemma StrictAntiOn.image_Ioi_subset (h : StrictAntiOn f (Ici a)) : f '' Ioi a ⊆ Iio (f a) 
goal: f '' Set.Ioi a ⊆ Set.Iio (f a)

full_name: StrictAntiOn.image_Iio_subset
code: lemma StrictAntiOn.image_Iio_subset (h : StrictAntiOn f (Iic b)) : f '' Iio b ⊆ Ioi (f b) 
goal: f '' Set.Iio b ⊆ Set.Ioi (f b)

full_name: StrictAntiOn.image_Ioo_subset
code: lemma StrictAntiOn.image_Ioo_subset (h : StrictAntiOn f (Icc a b)) :
    f '' Ioo a b ⊆ Ioo (f b) (f a) 
goal: f '' Set.Ioo a b ⊆ Set.Ioo (f b) (f a)

full_name: Monotone.image_Ici_subset
code: lemma Monotone.image_Ici_subset (h : Monotone f) : f '' Ici a ⊆ Ici (f a) 
goal: f '' Set.Ici a ⊆ Set.Ici (f a)

full_name: Monotone.image_Iic_subset
code: lemma Monotone.image_Iic_subset (h : Monotone f) : f '' Iic b ⊆ Iic (f b) 
goal: f '' Set.Iic b ⊆ Set.Iic (f b)

full_name: Monotone.image_Icc_subset
code: lemma Monotone.image_Icc_subset (h : Monotone f) : f '' Icc a b ⊆ Icc (f a) (f b) 
goal: f '' Set.Icc a b ⊆ Set.Icc (f a) (f b)

full_name: Antitone.image_Ici_subset
code: lemma Antitone.image_Ici_subset (h : Antitone f) : f '' Ici a ⊆ Iic (f a) 
goal: f '' Set.Ici a ⊆ Set.Iic (f a)

full_name: Antitone.image_Iic_subset
code: lemma Antitone.image_Iic_subset (h : Antitone f) : f '' Iic b ⊆ Ici (f b) 
goal: f '' Set.Iic b ⊆ Set.Ici (f b)

full_name: Antitone.image_Icc_subset
code: lemma Antitone.image_Icc_subset (h : Antitone f) : f '' Icc a b ⊆ Icc (f b) (f a) 
goal: f '' Set.Icc a b ⊆ Set.Icc (f b) (f a)

full_name: StrictMono.image_Ioi_subset
code: lemma StrictMono.image_Ioi_subset (h : StrictMono f) : f '' Ioi a ⊆ Ioi (f a) 
goal: f '' Set.Ioi a ⊆ Set.Ioi (f a)

full_name: StrictMono.image_Iio_subset
code: lemma StrictMono.image_Iio_subset (h : StrictMono f) : f '' Iio b ⊆ Iio (f b) 
goal: f '' Set.Iio b ⊆ Set.Iio (f b)

full_name: StrictMono.image_Ioo_subset
code: lemma StrictMono.image_Ioo_subset (h : StrictMono f) : f '' Ioo a b ⊆ Ioo (f a) (f b) 
goal: f '' Set.Ioo a b ⊆ Set.Ioo (f a) (f b)

full_name: StrictAnti.image_Ioi_subset
code: lemma StrictAnti.image_Ioi_subset (h : StrictAnti f) : f '' Ioi a ⊆ Iio (f a) 
goal: f '' Set.Ioi a ⊆ Set.Iio (f a)

full_name: StrictAnti.image_Iio_subset
code: lemma StrictAnti.image_Iio_subset (h : StrictAnti f) : f '' Iio b ⊆ Ioi (f b) 
goal: f '' Set.Iio b ⊆ Set.Ioi (f b)

full_name: StrictAnti.image_Ioo_subset
code: lemma StrictAnti.image_Ioo_subset (h : StrictAnti f) : f '' Ioo a b ⊆ Ioo (f b) (f a) 
goal: f '' Set.Ioo a b ⊆ Set.Ioo (f b) (f a)

full_name: StrictMonoOn.mapsTo_Ico
code: lemma StrictMonoOn.mapsTo_Ico (h : StrictMonoOn f (Icc a b)) :
    MapsTo f (Ico a b) (Ico (f a) (f b)) 
goal: Set.MapsTo f (Set.Ico a b) (Set.Ico (f a) (f b))

full_name: StrictMonoOn.mapsTo_Ioc
code: lemma StrictMonoOn.mapsTo_Ioc (h : StrictMonoOn f (Icc a b)) :
    MapsTo f (Ioc a b) (Ioc (f a) (f b)) 
goal: Set.MapsTo f (Set.Ioc a b) (Set.Ioc (f a) (f b))

full_name: StrictAntiOn.mapsTo_Ico
code: lemma StrictAntiOn.mapsTo_Ico (h : StrictAntiOn f (Icc a b)) :
    MapsTo f (Ico a b) (Ioc (f b) (f a)) 
goal: Set.MapsTo f (Set.Ico a b) (Set.Ioc (f b) (f a))

full_name: StrictAntiOn.mapsTo_Ioc
code: lemma StrictAntiOn.mapsTo_Ioc (h : StrictAntiOn f (Icc a b)) :
    MapsTo f (Ioc a b) (Ico (f b) (f a)) 
goal: Set.MapsTo f (Set.Ioc a b) (Set.Ico (f b) (f a))

full_name: StrictMono.mapsTo_Ico
code: lemma StrictMono.mapsTo_Ico (h : StrictMono f) : MapsTo f (Ico a b) (Ico (f a) (f b)) 
goal: Set.MapsTo f (Set.Ico a b) (Set.Ico (f a) (f b))

full_name: StrictMono.mapsTo_Ioc
code: lemma StrictMono.mapsTo_Ioc (h : StrictMono f) : MapsTo f (Ioc a b) (Ioc (f a) (f b)) 
goal: Set.MapsTo f (Set.Ioc a b) (Set.Ioc (f a) (f b))

full_name: StrictAnti.mapsTo_Ico
code: lemma StrictAnti.mapsTo_Ico (h : StrictAnti f) : MapsTo f (Ico a b) (Ioc (f b) (f a)) 
goal: Set.MapsTo f (Set.Ico a b) (Set.Ioc (f b) (f a))

full_name: StrictAnti.mapsTo_Ioc
code: lemma StrictAnti.mapsTo_Ioc (h : StrictAnti f) : MapsTo f (Ioc a b) (Ico (f b) (f a)) 
goal: Set.MapsTo f (Set.Ioc a b) (Set.Ico (f b) (f a))

full_name: StrictMonoOn.image_Ico_subset
code: lemma StrictMonoOn.image_Ico_subset (h : StrictMonoOn f (Icc a b)) :
    f '' Ico a b ⊆ Ico (f a) (f b) 
goal: f '' Set.Ico a b ⊆ Set.Ico (f a) (f b)

full_name: StrictMonoOn.image_Ioc_subset
code: lemma StrictMonoOn.image_Ioc_subset (h : StrictMonoOn f (Icc a b)) :
    f '' Ioc a b ⊆ Ioc (f a) (f b) 
goal: f '' Set.Ioc a b ⊆ Set.Ioc (f a) (f b)

full_name: StrictAntiOn.image_Ico_subset
code: lemma StrictAntiOn.image_Ico_subset (h : StrictAntiOn f (Icc a b)) :
    f '' Ico a b ⊆ Ioc (f b) (f a) 
goal: f '' Set.Ico a b ⊆ Set.Ioc (f b) (f a)

full_name: StrictAntiOn.image_Ioc_subset
code: lemma StrictAntiOn.image_Ioc_subset (h : StrictAntiOn f (Icc a b)) :
    f '' Ioc a b ⊆ Ico (f b) (f a) 
goal: f '' Set.Ioc a b ⊆ Set.Ico (f b) (f a)

full_name: StrictMono.image_Ico_subset
code: lemma StrictMono.image_Ico_subset (h : StrictMono f) : f '' Ico a b ⊆ Ico (f a) (f b) 
goal: f '' Set.Ico a b ⊆ Set.Ico (f a) (f b)

full_name: StrictMono.image_Ioc_subset
code: lemma StrictMono.image_Ioc_subset (h : StrictMono f) : f '' Ioc a b ⊆ Ioc (f a) (f b) 
goal: f '' Set.Ioc a b ⊆ Set.Ioc (f a) (f b)

full_name: StrictAnti.image_Ico_subset
code: lemma StrictAnti.image_Ico_subset (h : StrictAnti f) : f '' Ico a b ⊆ Ioc (f b) (f a) 
goal: f '' Set.Ico a b ⊆ Set.Ioc (f b) (f a)

full_name: StrictAnti.image_Ioc_subset
code: lemma StrictAnti.image_Ioc_subset (h : StrictAnti f) : f '' Ioc a b ⊆ Ico (f b) (f a) 
goal: f '' Set.Ioc a b ⊆ Set.Ico (f b) (f a)

full_name: _private.Mathlib.Order.Interval.Set.Image.0.Set.image_subtype_val_Ixx_Ixi
code: 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} 
goal: Subtype.val '' {y | r ↑c ↑y} = {y | r (↑c) y ∧ q y b}

full_name: _private.Mathlib.Order.Interval.Set.Image.0.Set.image_subtype_val_Ixx_Iix
code: 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} 
goal: Subtype.val '' {y | r ↑y ↑c} = {y | p a y ∧ r y ↑c}

full_name: Set.preimage_subtype_val_Ici
code: @[simp] lemma preimage_subtype_val_Ici (a : {x // p x}) : (↑) ⁻¹' (Ici a.1) = Ici a 
goal: Subtype.val ⁻¹' Set.Ici ↑a = Set.Ici a

full_name: Set.preimage_subtype_val_Iic
code: @[simp] lemma preimage_subtype_val_Iic (a : {x // p x}) : (↑) ⁻¹' (Iic a.1) = Iic a 
goal: Subtype.val ⁻¹' Set.Iic ↑a = Set.Iic a

full_name: Set.preimage_subtype_val_Ioi
code: @[simp] lemma preimage_subtype_val_Ioi (a : {x // p x}) : (↑) ⁻¹' (Ioi a.1) = Ioi a 
goal: Subtype.val ⁻¹' Set.Ioi ↑a = Set.Ioi a

full_name: Set.preimage_subtype_val_Iio
code: @[simp] lemma preimage_subtype_val_Iio (a : {x // p x}) : (↑) ⁻¹' (Iio a.1) = Iio a 
goal: Subtype.val ⁻¹' Set.Iio ↑a = Set.Iio a

full_name: Set.preimage_subtype_val_Icc
code: @[simp] lemma preimage_subtype_val_Icc (a b : {x // p x}) : (↑) ⁻¹' (Icc a.1 b) = Icc a b 
goal: Subtype.val ⁻¹' Set.Icc ↑a ↑b = Set.Icc a b

full_name: Set.preimage_subtype_val_Ico
code: @[simp] lemma preimage_subtype_val_Ico (a b : {x // p x}) : (↑) ⁻¹' (Ico a.1 b) = Ico a b 
goal: Subtype.val ⁻¹' Set.Ico ↑a ↑b = Set.Ico a b

full_name: Set.preimage_subtype_val_Ioc
code: @[simp] lemma preimage_subtype_val_Ioc (a b : {x // p x}) : (↑) ⁻¹' (Ioc a.1 b) = Ioc a b 
goal: Subtype.val ⁻¹' Set.Ioc ↑a ↑b = Set.Ioc a b

full_name: Set.preimage_subtype_val_Ioo
code: @[simp] lemma preimage_subtype_val_Ioo (a b : {x // p x}) : (↑) ⁻¹' (Ioo a.1 b) = Ioo a b 
goal: Subtype.val ⁻¹' Set.Ioo ↑a ↑b = Set.Ioo a b

full_name: Set.image_subtype_val_Icc_subset
code: theorem image_subtype_val_Icc_subset (a b : {x // p x}) :
    Subtype.val '' Icc a b ⊆ Icc a.val b.val 
goal: Subtype.val '' Set.Icc a b ⊆ Set.Icc ↑a ↑b

full_name: Set.image_subtype_val_Ico_subset
code: theorem image_subtype_val_Ico_subset (a b : {x // p x}) :
    Subtype.val '' Ico a b ⊆ Ico a.val b.val 
goal: Subtype.val '' Set.Ico a b ⊆ Set.Ico ↑a ↑b

full_name: Set.image_subtype_val_Ioc_subset
code: theorem image_subtype_val_Ioc_subset (a b : {x // p x}) :
    Subtype.val '' Ioc a b ⊆ Ioc a.val b.val 
goal: Subtype.val '' Set.Ioc a b ⊆ Set.Ioc ↑a ↑b

full_name: Set.image_subtype_val_Ioo_subset
code: theorem image_subtype_val_Ioo_subset (a b : {x // p x}) :
    Subtype.val '' Ioo a b ⊆ Ioo a.val b.val 
goal: Subtype.val '' Set.Ioo a b ⊆ Set.Ioo ↑a ↑b

full_name: Set.image_subtype_val_Iic_subset
code: theorem image_subtype_val_Iic_subset (a : {x // p x}) :
    Subtype.val '' Iic a ⊆ Iic a.val 
goal: Subtype.val '' Set.Iic a ⊆ Set.Iic ↑a

full_name: Set.image_subtype_val_Iio_subset
code: theorem image_subtype_val_Iio_subset (a : {x // p x}) :
    Subtype.val '' Iio a ⊆ Iio a.val 
goal: Subtype.val '' Set.Iio a ⊆ Set.Iio ↑a

full_name: Set.image_subtype_val_Ici_subset
code: theorem image_subtype_val_Ici_subset (a : {x // p x}) :
    Subtype.val '' Ici a ⊆ Ici a.val 
goal: Subtype.val '' Set.Ici a ⊆ Set.Ici ↑a

full_name: Set.image_subtype_val_Ioi_subset
code: theorem image_subtype_val_Ioi_subset (a : {x // p x}) :
    Subtype.val '' Ioi a ⊆ Ioi a.val 
goal: Subtype.val '' Set.Ioi a ⊆ Set.Ioi ↑a

full_name: Set.image_subtype_val_Ici_Iic
code: @[simp]
lemma image_subtype_val_Ici_Iic {a : α} (b : Ici a) : Subtype.val '' Iic b = Icc a b 
goal: Subtype.val '' Set.Iic b = Set.Icc a ↑b

full_name: Set.image_subtype_val_Ici_Iio
code: @[simp]
lemma image_subtype_val_Ici_Iio {a : α} (b : Ici a) : Subtype.val '' Iio b = Ico a b 
goal: Subtype.val '' Set.Iio b = Set.Ico a ↑b

full_name: Set.image_subtype_val_Ici_Ici
code: @[simp]
lemma image_subtype_val_Ici_Ici {a : α} (b : Ici a) : Subtype.val '' Ici b = Ici b.1 
goal: Subtype.val '' Set.Ici b = Set.Ici ↑b

full_name: Set.image_subtype_val_Ici_Ioi
code: @[simp]
lemma image_subtype_val_Ici_Ioi {a : α} (b : Ici a) : Subtype.val '' Ioi b = Ioi b.1 
goal: Subtype.val '' Set.Ioi b = Set.Ioi ↑b

full_name: Set.image_subtype_val_Iic_Ici
code: @[simp]
lemma image_subtype_val_Iic_Ici {a : α} (b : Iic a) : Subtype.val '' Ici b = Icc b.1 a 
goal: Subtype.val '' Set.Ici b = Set.Icc (↑b) a

full_name: Set.image_subtype_val_Iic_Ioi
code: @[simp]
lemma image_subtype_val_Iic_Ioi {a : α} (b : Iic a) : Subtype.val '' Ioi b = Ioc b.1 a 
goal: Subtype.val '' Set.Ioi b = Set.Ioc (↑b) a

full_name: Set.image_subtype_val_Iic_Iic
code: @[simp]
lemma image_subtype_val_Iic_Iic {a : α} (b : Iic a) : Subtype.val '' Iic b = Iic b.1 
goal: Subtype.val '' Set.Iic b = Set.Iic ↑b

full_name: Set.image_subtype_val_Iic_Iio
code: @[simp]
lemma image_subtype_val_Iic_Iio {a : α} (b : Iic a) : Subtype.val '' Iio b = Iio b.1 
goal: Subtype.val '' Set.Iio b = Set.Iio ↑b

full_name: Set.image_subtype_val_Ioi_Ici
code: @[simp]
lemma image_subtype_val_Ioi_Ici {a : α} (b : Ioi a) : Subtype.val '' Ici b = Ici b.1 
goal: Subtype.val '' Set.Ici b = Set.Ici ↑b

full_name: Set.image_subtype_val_Ioi_Iic
code: @[simp]
lemma image_subtype_val_Ioi_Iic {a : α} (b : Ioi a) : Subtype.val '' Iic b = Ioc a b 
goal: Subtype.val '' Set.Iic b = Set.Ioc a ↑b

full_name: Set.image_subtype_val_Ioi_Ioi
code: @[simp]
lemma image_subtype_val_Ioi_Ioi {a : α} (b : Ioi a) : Subtype.val '' Ioi b = Ioi b.1 
goal: Subtype.val '' Set.Ioi b = Set.Ioi ↑b

full_name: Set.image_subtype_val_Ioi_Iio
code: @[simp]
lemma image_subtype_val_Ioi_Iio {a : α} (b : Ioi a) : Subtype.val '' Iio b = Ioo a b 
goal: Subtype.val '' Set.Iio b = Set.Ioo a ↑b

full_name: Set.image_subtype_val_Iio_Ici
code: @[simp]
lemma image_subtype_val_Iio_Ici {a : α} (b : Iio a) : Subtype.val '' Ici b = Ico b.1 a 
goal: Subtype.val '' Set.Ici b = Set.Ico (↑b) a

full_name: Set.image_subtype_val_Iio_Iic
code: @[simp]
lemma image_subtype_val_Iio_Iic {a : α} (b : Iio a) : Subtype.val '' Iic b = Iic b.1 
goal: Subtype.val '' Set.Iic b = Set.Iic ↑b

full_name: Set.image_subtype_val_Iio_Ioi
code: @[simp]
lemma image_subtype_val_Iio_Ioi {a : α} (b : Iio a) : Subtype.val '' Ioi b = Ioo b.1 a 
goal: Subtype.val '' Set.Ioi b = Set.Ioo (↑b) a

full_name: Set.image_subtype_val_Iio_Iio
code: @[simp]
lemma image_subtype_val_Iio_Iio {a : α} (b : Iio a) : Subtype.val '' Iio b = Iio b.1 
goal: Subtype.val '' Set.Iio b = Set.Iio ↑b

full_name: Set.image_subtype_val_Icc_Ici
code: @[simp]
lemma image_subtype_val_Icc_Ici {a b : α} (c : Icc a b) : Subtype.val '' Ici c = Icc c.1 b 
goal: Subtype.val '' Set.Ici c = Set.Icc (↑c) b

full_name: Set.image_subtype_val_Icc_Iic
code: @[simp]
lemma image_subtype_val_Icc_Iic {a b : α} (c : Icc a b) : Subtype.val '' Iic c = Icc a c 
goal: Subtype.val '' Set.Iic c = Set.Icc a ↑c

full_name: Set.image_subtype_val_Icc_Ioi
code: @[simp]
lemma image_subtype_val_Icc_Ioi {a b : α} (c : Icc a b) : Subtype.val '' Ioi c = Ioc c.1 b 
goal: Subtype.val '' Set.Ioi c = Set.Ioc (↑c) b

full_name: Set.image_subtype_val_Icc_Iio
code: @[simp]
lemma image_subtype_val_Icc_Iio {a b : α} (c : Icc a b) : Subtype.val '' Iio c = Ico a c 
goal: Subtype.val '' Set.Iio c = Set.Ico a ↑c

full_name: Set.image_subtype_val_Ico_Ici
code: @[simp]
lemma image_subtype_val_Ico_Ici {a b : α} (c : Ico a b) : Subtype.val '' Ici c = Ico c.1 b 
goal: Subtype.val '' Set.Ici c = Set.Ico (↑c) b

full_name: Set.image_subtype_val_Ico_Iic
code: @[simp]
lemma image_subtype_val_Ico_Iic {a b : α} (c : Ico a b) : Subtype.val '' Iic c = Icc a c 
goal: Subtype.val '' Set.Iic c = Set.Icc a ↑c

full_name: Set.image_subtype_val_Ico_Ioi
code: @[simp]
lemma image_subtype_val_Ico_Ioi {a b : α} (c : Ico a b) : Subtype.val '' Ioi c = Ioo c.1 b 
goal: Subtype.val '' Set.Ioi c = Set.Ioo (↑c) b

full_name: Set.image_subtype_val_Ico_Iio
code: @[simp]
lemma image_subtype_val_Ico_Iio {a b : α} (c : Ico a b) : Subtype.val '' Iio c = Ico a c 
goal: Subtype.val '' Set.Iio c = Set.Ico a ↑c

full_name: Set.image_subtype_val_Ioc_Ici
code: @[simp]
lemma image_subtype_val_Ioc_Ici {a b : α} (c : Ioc a b) : Subtype.val '' Ici c = Icc c.1 b 
goal: Subtype.val '' Set.Ici c = Set.Icc (↑c) b

full_name: Set.image_subtype_val_Ioc_Iic
code: @[simp]
lemma image_subtype_val_Ioc_Iic {a b : α} (c : Ioc a b) : Subtype.val '' Iic c = Ioc a c 
goal: Subtype.val '' Set.Iic c = Set.Ioc a ↑c

full_name: Set.image_subtype_val_Ioc_Ioi
code: @[simp]
lemma image_subtype_val_Ioc_Ioi {a b : α} (c : Ioc a b) : Subtype.val '' Ioi c = Ioc c.1 b 
goal: Subtype.val '' Set.Ioi c = Set.Ioc (↑c) b

full_name: Set.image_subtype_val_Ioc_Iio
code: @[simp]
lemma image_subtype_val_Ioc_Iio {a b : α} (c : Ioc a b) : Subtype.val '' Iio c = Ioo a c 
goal: Subtype.val '' Set.Iio c = Set.Ioo a ↑c

full_name: Set.image_subtype_val_Ioo_Ici
code: @[simp]
lemma image_subtype_val_Ioo_Ici {a b : α} (c : Ioo a b) : Subtype.val '' Ici c = Ico c.1 b 
goal: Subtype.val '' Set.Ici c = Set.Ico (↑c) b

full_name: Set.image_subtype_val_Ioo_Iic
code: @[simp]
lemma image_subtype_val_Ioo_Iic {a b : α} (c : Ioo a b) : Subtype.val '' Iic c = Ioc a c 
goal: Subtype.val '' Set.Iic c = Set.Ioc a ↑c

full_name: Set.image_subtype_val_Ioo_Ioi
code: @[simp]
lemma image_subtype_val_Ioo_Ioi {a b : α} (c : Ioo a b) : Subtype.val '' Ioi c = Ioo c.1 b 
goal: Subtype.val '' Set.Ioi c = Set.Ioo (↑c) b

full_name: Set.image_subtype_val_Ioo_Iio
code: @[simp]
lemma image_subtype_val_Ioo_Iio {a b : α} (c : Ioo a b) : Subtype.val '' Iio c = Ioo a c 
goal: Subtype.val '' Set.Iio c = Set.Ioo a ↑c

full_name: directedOn_le_Iic
code: lemma directedOn_le_Iic (b : α) : DirectedOn (· ≤ ·) (Iic b) 
goal: DirectedOn (fun x x_1 => x ≤ x_1) (Set.Iic b)

full_name: directedOn_le_Icc
code: lemma directedOn_le_Icc (a b : α) : DirectedOn (· ≤ ·) (Icc a b) 
goal: DirectedOn (fun x x_1 => x ≤ x_1) (Set.Icc a b)

full_name: directedOn_le_Ioc
code: lemma directedOn_le_Ioc (a b : α) : DirectedOn (· ≤ ·) (Ioc a b) 
goal: DirectedOn (fun x x_1 => x ≤ x_1) (Set.Ioc a b)

full_name: directedOn_ge_Ici
code: lemma directedOn_ge_Ici (a : α) : DirectedOn (· ≥ ·) (Ici a) 
goal: DirectedOn (fun x x_1 => x ≥ x_1) (Set.Ici a)

full_name: directedOn_ge_Icc
code: lemma directedOn_ge_Icc (a b : α) : DirectedOn (· ≥ ·) (Icc a b) 
goal: DirectedOn (fun x x_1 => x ≥ x_1) (Set.Icc a b)

full_name: directedOn_ge_Ico
code: lemma directedOn_ge_Ico (a b : α) : DirectedOn (· ≥ ·) (Ico a b) 
goal: DirectedOn (fun x x_1 => x ≥ x_1) (Set.Ico a b)