Documentation Verification Report

Map

πŸ“ Source: Mathlib/Order/Filter/Map.lean

Statistics

MetricCount
DefinitionsinstAlternative, kernMap, monad, Map, Map
5
Theoremscomap, comap_of_image_mem, comap_of_range_mem, comap_of_surj, map, of_map, biSup_pure_eq_principal, bind_def, bind_inf_principal, bind_le, bind_map, bind_mono, canLift, comap_bot, comap_coe_neBot_of_le_principal, comap_comap, comap_comm, comap_const_of_mem, comap_const_of_notMem, comap_eq_bot_iff_compl_range, comap_equiv_symm, comap_eval_neBot, comap_eval_neBot_iff, comap_eval_neBot_iff', comap_fst_neBot, comap_fst_neBot_iff, comap_iInf, comap_iSup, comap_id, comap_id', comap_inf, comap_inf_principal_neBot_of_image_mem, comap_inf_principal_range, comap_injective, comap_inl_map_inr, comap_inr_map_inl, comap_le_comap_iff, comap_le_iff_le_kernMap, comap_map, comap_mono, comap_neBot, comap_neBot_iff, comap_neBot_iff_compl_range, comap_neBot_iff_frequently, comap_principal, comap_pure, comap_sSup, comap_snd_neBot, comap_snd_neBot_iff, comap_sumElim_eq, comap_sup, comap_surjective_eq_bot, comap_top, compl_mem_comap, compl_mem_kernMap, disjoint_comap, disjoint_comap_iff, disjoint_comap_iff_map, disjoint_comap_iff_map', disjoint_map, disjoint_of_map, eventuallyEq_bind, eventuallyEq_map, eventuallyLE_bind, eventuallyLE_map, eventually_bind, eventually_comap, eventually_map, eventually_pure, filter_injOn_Iic_iff_injOn, frequently_bind, frequently_comap, frequently_map, gc_comap_kernMap, gc_map_comap, generate_image_preimage_le_comap, iSup_pure_eq_top, image_coe_mem_of_mem_comap, image_mem_map, image_mem_map_iff, image_mem_of_mem_comap, inf_principal_eq_bot_iff_comap, instCommApplicative, instLawfulApplicative, instLawfulFunctor, join_pure, kernMap_principal, lawfulMonad, le_comap_map, le_comap_top, le_map, le_map_iff, le_pure_iff, le_seq, map_biInf_eq, map_bind, map_bot, map_comap, map_comap_inl_sup_map_comap_inr, map_comap_le, map_comap_of_mem, map_comap_of_surjective, map_comap_setCoe_val, map_comm, map_compose, map_congr, map_const, map_def, map_eq_bot_iff, map_eq_comap_of_inverse, map_eq_map_iff_of_injOn, map_equiv_symm, map_generate_le_generate_preimage_preimage, map_iInf_eq, map_iInf_le, map_iSup, map_id, map_id', map_inf, map_inf', map_inf_le, map_inj, map_injective, map_inl_inf_map_inr, map_inr_inf_map_inl, map_le_iff_le_comap, map_le_map_iff, map_le_map_iff_of_injOn, map_map, map_mono, map_neBot, map_neBot_iff, map_principal, map_pure, map_sumElim_eq, map_sup, map_surjOn_Iic_iff_le_map, map_surjOn_Iic_iff_surjOn, map_swap4_eq_comap, map_swap_eq_comap_swap, map_top, mem_bind, mem_bind', mem_comap, mem_comap', mem_comap'', mem_comap_iff, mem_comap_iff_compl, mem_comap_prodMk, mem_kernMap, mem_kernMap_iff_compl, mem_map, mem_map', mem_map_iff_exists_image, mem_map_seq_iff, mem_seq_def, mem_seq_iff, neBot_inf_comap_iff_map, neBot_inf_comap_iff_map', neBot_of_comap, preimage_mem_comap, principal_bind, principal_eq_map_coe_top, principal_singleton, principal_subtype, prod_map_seq_comm, pure_bind, pure_injective, pure_le_principal, pure_neBot, pure_seq_eq_map, pure_sets, push_pull, push_pull', range_mem_map, sInter_comap_sets, seq_assoc, seq_eq_filter_seq, seq_mem_seq, seq_mono, seq_pure, singleton_mem_pure, subtype_coe_map_comap, sup_bind, filter_comap, filter_map, filter_comap, filter_map, filter_comap, filter_map, filter_comap, filter_map, filter_map_top, filter_map_Iic, filter_map_Iic, filter_map_Iic, filter_map_Iic
197
Total202

Filter

Definitions

NameCategoryTheorems
instAlternative πŸ“–CompOp
8 mathmath: mem_traverse_iff, nhds_cons, instLawfulApplicative, instCommApplicative, sequence_mono, mem_traverse, seq_eq_filter_seq, nhds_list
kernMap πŸ“–CompOp
6 mathmath: compl_mem_kernMap, kernMap_principal, gc_comap_kernMap, mem_kernMap_iff_compl, comap_le_iff_le_kernMap, mem_kernMap
monad πŸ“–CompOp
1 mathmath: lawfulMonad

Theorems

NameKindAssumesProvesValidatesDepends On
biSup_pure_eq_principal πŸ“–mathematicalβ€”iSup
Filter
instSupSet
Set
Set.instMembership
instPure
principal
β€”ext
bind_def πŸ“–mathematicalβ€”Filter
instBind
bind
β€”β€”
bind_inf_principal πŸ“–mathematicalβ€”bind
Filter
instInf
principal
β€”ext
bind_le πŸ“–mathematicalEventually
Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
bindβ€”join_le
eventually_map
bind_map πŸ“–mathematicalβ€”bind
map
Filter
β€”β€”
bind_mono πŸ“–mathematicalFilter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
EventuallyLE
bindβ€”le_trans
mp_mem
univ_mem'
join_mono
map_mono
canLift πŸ“–mathematicalβ€”CanLift
Filter
map
Eventually
β€”map_comap_of_mem
Eventually.mono
CanLift.prf
comap_bot πŸ“–mathematicalβ€”comap
Bot.bot
Filter
instBot
β€”bot_unique
mem_bot
comap_coe_neBot_of_le_principal πŸ“–mathematicalFilter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
principal
NeBot
Set
Set.instMembership
comap
β€”NeBot.comap_of_range_mem
mem_principal_self
Subtype.range_coe
comap_comap πŸ“–mathematicalβ€”comapβ€”coext
Set.image_image
Set.image_congr
comap_comm πŸ“–mathematicalβ€”comapβ€”comap_comap
comap_const_of_mem πŸ“–mathematicalSet
Set.instMembership
comap
Top.top
Filter
instTop
β€”top_unique
univ_mem'
mem_comap'
comap_const_of_notMem πŸ“–mathematicalSet
Filter
instMembership
Set.instMembership
comap
Bot.bot
instBot
β€”empty_mem_iff_bot
mem_comap'
mem_of_superset
comap_eq_bot_iff_compl_range πŸ“–mathematicalβ€”comap
Bot.bot
Filter
instBot
Set
instMembership
Compl.compl
Set.instCompl
Set.range
β€”not_iff_not
neBot_iff
comap_neBot_iff_compl_range
comap_equiv_symm πŸ“–mathematicalβ€”comap
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
map
β€”map_eq_comap_of_inverse
Equiv.self_comp_symm
Equiv.symm_comp_self
comap_eval_neBot πŸ“–mathematicalβ€”NeBot
comap
Function.eval
β€”comap_eval_neBot_iff
comap_eval_neBot_iff πŸ“–mathematicalβ€”NeBot
comap
Function.eval
β€”β€”
comap_eval_neBot_iff' πŸ“–mathematicalβ€”NeBot
comap
Function.eval
β€”isEmpty_or_nonempty
filter_eq_bot_of_isEmpty
not_iff_not
Classical.nonempty_pi
Set.range_eval
comap_fst_neBot πŸ“–mathematicalβ€”NeBot
comap
β€”comap_fst_neBot_iff
comap_fst_neBot_iff πŸ“–mathematicalβ€”NeBot
comap
β€”isEmpty_or_nonempty
filter_eq_bot_of_isEmpty
Prod.isEmpty_right
not_iff_not
Prod.range_fst
comap_iInf πŸ“–mathematicalβ€”comap
iInf
Filter
instInfSet
β€”GaloisConnection.u_iInf
gc_map_comap
comap_iSup πŸ“–mathematicalβ€”comap
iSup
Filter
instSupSet
β€”GaloisConnection.l_iSup
gc_comap_kernMap
comap_id πŸ“–mathematicalβ€”comapβ€”le_antisymm
preimage_mem_comap
mem_of_superset
comap_id' πŸ“–mathematicalβ€”comapβ€”comap_id
comap_inf πŸ“–mathematicalβ€”comap
Filter
instInf
β€”GaloisConnection.u_inf
gc_map_comap
comap_inf_principal_neBot_of_image_mem πŸ“–mathematicalSet
Filter
instMembership
Set.image
NeBot
instInf
comap
principal
β€”neBot_inf_comap_iff_map'
map_principal
frequently_mem_iff_neBot
Eventually.frequently
comap_inf_principal_range πŸ“–mathematicalβ€”comap
Filter
instInf
principal
Set.range
β€”comap_inf
comap_principal
Set.preimage_range
principal_univ
inf_of_le_left
comap_injective πŸ“–mathematicalβ€”Filter
comap
β€”map_comap_of_surjective
comap_inl_map_inr πŸ“–mathematicalβ€”comap
map
Bot.bot
Filter
instBot
β€”ext
mem_comap_iff_compl
Set.preimage_inr_image_inl
Set.compl_empty
comap_inr_map_inl πŸ“–mathematicalβ€”comap
map
Bot.bot
Filter
instBot
β€”ext
mem_comap_iff_compl
Set.preimage_inl_image_inr
Set.compl_empty
comap_le_comap_iff πŸ“–mathematicalSet
Filter
instMembership
Set.range
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
β€”LE.le.trans
map_mono
map_comap_le
map_comap_of_mem
comap_mono
comap_le_iff_le_kernMap πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
kernMap
β€”β€”
comap_map πŸ“–mathematicalβ€”comap
map
β€”le_antisymm
mem_of_superset
preimage_mem_comap
image_mem_map
Set.preimage_image_eq
le_comap_map
comap_mono πŸ“–mathematicalβ€”Monotone
Filter
PartialOrder.toPreorder
instPartialOrder
comap
β€”GaloisConnection.monotone_u
gc_map_comap
comap_neBot πŸ“–mathematicalSet
Set.instMembership
NeBot
comap
β€”comap_neBot_iff
comap_neBot_iff πŸ“–mathematicalβ€”NeBot
comap
Set
Set.instMembership
β€”Set.Subset.rfl
comap_neBot_iff_compl_range πŸ“–mathematicalβ€”NeBot
comap
Set
Filter
instMembership
Compl.compl
Set.instCompl
Set.range
β€”comap_neBot_iff_frequently
comap_neBot_iff_frequently πŸ“–mathematicalβ€”NeBot
comap
Frequently
Set
Set.instMembership
Set.range
β€”β€”
comap_principal πŸ“–mathematicalβ€”comap
principal
Set.preimage
β€”ext
HasSubset.Subset.trans
Set.instIsTransSubset
Set.preimage_mono
Set.Subset.rfl
comap_pure πŸ“–mathematicalβ€”comap
Filter
instPure
principal
Set.preimage
Set
Set.instSingletonSet
β€”principal_singleton
comap_principal
comap_sSup πŸ“–mathematicalβ€”comap
SupSet.sSup
Filter
instSupSet
iSup
Set
Set.instMembership
β€”sSup_eq_iSup
comap_iSup
comap_snd_neBot πŸ“–mathematicalβ€”NeBot
comap
β€”comap_snd_neBot_iff
comap_snd_neBot_iff πŸ“–mathematicalβ€”NeBot
comap
β€”isEmpty_or_nonempty
filter_eq_bot_of_isEmpty
Prod.isEmpty_left
not_iff_not
Prod.range_snd
comap_sumElim_eq πŸ“–mathematicalβ€”comap
Filter
instSup
map
β€”ext
Set.image_sumElim
Set.compl_union
comap_sup πŸ“–mathematicalβ€”comap
Filter
instSup
β€”sup_eq_iSup
comap_iSup
iSup_bool_eq
comap_surjective_eq_bot πŸ“–mathematicalβ€”comap
Bot.bot
Filter
instBot
β€”comap_eq_bot_iff_compl_range
Function.Surjective.range_eq
Set.compl_univ
empty_mem_iff_bot
comap_top πŸ“–mathematicalβ€”comap
Top.top
Filter
instTop
β€”GaloisConnection.u_top
gc_map_comap
compl_mem_comap πŸ“–mathematicalβ€”Set
Filter
instMembership
comap
Compl.compl
Set.instCompl
Set.image
β€”mem_comap_iff_compl
compl_compl
compl_mem_kernMap πŸ“–mathematicalβ€”Set
Filter
instMembership
kernMap
Compl.compl
Set.instCompl
Set.image
β€”compl_compl
disjoint_comap πŸ“–mathematicalDisjoint
Filter
instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
instCompleteLatticeFilter
comapβ€”Disjoint.eq_bot
comap_bot
disjoint_comap_iff πŸ“–mathematicalβ€”Disjoint
Filter
instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
instCompleteLatticeFilter
comap
β€”disjoint_iff
comap_inf
comap_surjective_eq_bot
disjoint_comap_iff_map πŸ“–mathematicalβ€”Disjoint
Filter
instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
instCompleteLatticeFilter
comap
map
β€”β€”
disjoint_comap_iff_map' πŸ“–mathematicalβ€”Disjoint
Filter
instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
instCompleteLatticeFilter
comap
map
β€”β€”
disjoint_map πŸ“–mathematicalβ€”Disjoint
Filter
instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
instCompleteLatticeFilter
map
β€”map_inf
disjoint_of_map πŸ“–β€”Disjoint
Filter
instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
instCompleteLatticeFilter
map
β€”β€”disjoint_iff
map_eq_bot_iff
le_bot_iff
map_inf_le
eventuallyEq_bind πŸ“–mathematicalβ€”EventuallyEq
bind
Eventually
β€”β€”
eventuallyEq_map πŸ“–mathematicalβ€”EventuallyEq
map
β€”β€”
eventuallyLE_bind πŸ“–mathematicalβ€”EventuallyLE
bind
Eventually
β€”β€”
eventuallyLE_map πŸ“–mathematicalβ€”EventuallyLE
map
β€”β€”
eventually_bind πŸ“–mathematicalβ€”Eventually
bind
β€”β€”
eventually_comap πŸ“–mathematicalβ€”Eventually
comap
β€”mem_comap'
eventually_map πŸ“–mathematicalβ€”Eventually
map
β€”β€”
eventually_pure πŸ“–mathematicalβ€”Eventually
Filter
instPure
β€”β€”
filter_injOn_Iic_iff_injOn πŸ“–mathematicalβ€”Set.InjOn
Filter
map
Set.Iic
PartialOrder.toPreorder
instPartialOrder
principal
β€”pure_injective
Set.InjOn.eq_iff
Set.mem_Iic
pure_le_principal
map_pure
map_eq_map_iff_of_injOn
le_principal_iff
frequently_bind πŸ“–mathematicalβ€”Frequently
bind
β€”not_iff_not
frequently_comap πŸ“–mathematicalβ€”Frequently
comap
β€”β€”
frequently_map πŸ“–mathematicalβ€”Frequently
map
β€”β€”
gc_comap_kernMap πŸ“–mathematicalβ€”GaloisConnection
Filter
PartialOrder.toPreorder
instPartialOrder
comap
kernMap
β€”comap_le_iff_le_kernMap
gc_map_comap πŸ“–mathematicalβ€”GaloisConnection
Filter
PartialOrder.toPreorder
instPartialOrder
map
comap
β€”map_le_iff_le_comap
generate_image_preimage_le_comap πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
generate
Set.image
Set
Set.preimage
comap
β€”map_le_iff_le_comap
le_generate_iff
mem_generate_of_mem
iSup_pure_eq_top πŸ“–mathematicalβ€”iSup
Filter
instSupSet
instPure
Top.top
instTop
β€”principal_univ
biSup_pure_eq_principal
iSup_univ
image_coe_mem_of_mem_comap πŸ“–mathematicalSet
Filter
instMembership
Set.Elem
Set.instMembership
comap
Set.imageβ€”image_mem_of_mem_comap
Subtype.range_coe_subtype
image_mem_map πŸ“–mathematicalSet
Filter
instMembership
map
Set.image
β€”sets_of_superset
Set.subset_preimage_image
image_mem_map_iff πŸ“–mathematicalβ€”Set
Filter
instMembership
map
Set.image
β€”Set.preimage_image_eq
image_mem_map
image_mem_of_mem_comap πŸ“–mathematicalSet
Filter
instMembership
Set.range
comap
Set.imageβ€”map_comap_of_mem
image_mem_map
inf_principal_eq_bot_iff_comap πŸ“–mathematicalβ€”Filter
instInf
principal
Bot.bot
instBot
comap
Set
Set.instMembership
β€”principal_eq_map_coe_top
push_pull'
inf_top_eq
map_eq_bot_iff
instCommApplicative πŸ“–mathematicalβ€”CommApplicative
Filter
instAlternative
β€”instLawfulApplicative
prod_map_seq_comm
instLawfulApplicative πŸ“–mathematicalβ€”Filter
instAlternative
β€”instLawfulFunctor
pure_seq_eq_map
map_pure
seq_pure
seq_assoc
instLawfulFunctor πŸ“–mathematicalβ€”Filter
instFunctor
β€”map_id
map_map
join_pure πŸ“–mathematicalβ€”join
Filter
instPure
β€”β€”
kernMap_principal πŸ“–mathematicalβ€”kernMap
principal
Set.kernImage
β€”eq_of_forall_le_iff
comap_le_iff_le_kernMap
le_principal_iff
mem_comap''
lawfulMonad πŸ“–mathematicalβ€”Filter
monad
β€”β€”
le_comap_map πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
map
β€”GaloisConnection.le_u_l
gc_map_comap
le_comap_top πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
comap
Top.top
instTop
β€”comap_top
le_top
le_map πŸ“–mathematicalSet
Filter
instMembership
Set.image
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
β€”mem_of_superset
Set.image_preimage_subset
le_map_iff πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
Set
instMembership
Set.image
β€”image_mem_map
le_map
le_pure_iff πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instPure
Set
instMembership
Set.instSingletonSet
β€”principal_singleton
le_principal_iff
le_seq πŸ“–mathematicalSet
Filter
instMembership
Set.seq
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
seq
β€”mem_of_superset
map_biInf_eq πŸ“–mathematicalDirectedOn
Order.Preimage
Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
setOf
map
iInf
instInfSet
β€”iInf_subtype'
map_iInf_eq
DirectedOn.directed_val
nonempty_subtype
map_bind πŸ“–mathematicalβ€”map
bind
Filter
β€”β€”
map_bot πŸ“–mathematicalβ€”map
Bot.bot
Filter
instBot
β€”GaloisConnection.l_bot
gc_map_comap
map_comap πŸ“–mathematicalβ€”map
comap
Filter
instInf
principal
Set.range
β€”le_antisymm
le_inf
map_comap_le
le_principal_iff
range_mem_map
mem_inf_principal
mem_of_superset
map_comap_inl_sup_map_comap_inr πŸ“–mathematicalβ€”Filter
instSup
map
comap
β€”comap_sumElim_eq
comap_id
map_comap_le πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
comap
β€”GaloisConnection.l_u_le
gc_map_comap
map_comap_of_mem πŸ“–mathematicalSet
Filter
instMembership
Set.range
map
comap
β€”map_comap
inf_eq_left
le_principal_iff
map_comap_of_surjective πŸ“–mathematicalβ€”map
comap
β€”map_comap_of_mem
Function.Surjective.range_eq
map_comap_setCoe_val πŸ“–mathematicalβ€”map
Set
Set.instMembership
comap
Filter
instInf
principal
β€”map_comap
Subtype.range_val
map_comm πŸ“–mathematicalβ€”mapβ€”map_map
map_compose πŸ“–mathematicalβ€”Filter
map
β€”filter_eq
map_congr πŸ“–mathematicalEventuallyEqmapβ€”ext'
eventually_congr
Eventually.mono
map_const πŸ“–mathematicalβ€”map
Filter
instPure
β€”ext
Set.preimage_const_of_mem
Set.preimage_const_of_notMem
map_def πŸ“–mathematicalβ€”Filter
instFunctor
map
β€”β€”
map_eq_bot_iff πŸ“–mathematicalβ€”map
Bot.bot
Filter
instBot
β€”empty_mem_iff_bot
map_bot
map_eq_comap_of_inverse πŸ“–mathematicalβ€”map
comap
β€”map_equiv_symm
map_eq_map_iff_of_injOn πŸ“–mathematicalSet
Filter
instMembership
Set.InjOn
mapβ€”map_le_map_iff_of_injOn
map_equiv_symm πŸ“–mathematicalβ€”map
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
comap
β€”map_injective
Equiv.injective
map_map
Equiv.self_comp_symm
map_id
map_comap_of_surjective
Equiv.surjective
map_generate_le_generate_preimage_preimage πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
generate
Set.preimage
Set
β€”le_generate_iff
mem_generate_of_mem
map_iInf_eq πŸ“–mathematicalDirected
Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
iInf
instInfSet
β€”LE.le.antisymm
map_iInf_le
mem_iInf_of_directed
iInf_le_of_le
le_principal_iff
map_iInf_le πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
iInf
instInfSet
β€”le_iInf
map_mono
iInf_le
map_iSup πŸ“–mathematicalβ€”map
iSup
Filter
instSupSet
β€”GaloisConnection.l_iSup
gc_map_comap
map_id πŸ“–mathematicalβ€”mapβ€”filter_eq
map_id' πŸ“–mathematicalβ€”mapβ€”map_id
map_inf πŸ“–mathematicalβ€”map
Filter
instInf
β€”LE.le.antisymm
map_inf_le
mem_inf_of_inter
image_mem_map
Set.image_inter
Set.image_subset_iff
subset_refl
Set.instReflSubset
map_inf' πŸ“–mathematicalSet
Filter
instMembership
Set.InjOn
map
instInf
β€”CanLift.prf
canLift
Subtype.canLift
Set.InjOn.injective
map_inf
Subtype.coe_injective
map_map
map_inf_le πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
instInf
β€”Monotone.map_inf_le
map_mono
map_inj πŸ“–mathematicalβ€”mapβ€”map_eq_map_iff_of_injOn
univ_mem
Function.Injective.injOn
map_injective πŸ“–mathematicalβ€”Filter
map
β€”β€”
map_inl_inf_map_inr πŸ“–mathematicalβ€”Filter
instInf
map
Bot.bot
instBot
β€”le_bot_iff
inf_le_inf
map_top
Set.preimage_range
inf_principal
Set.range_inl_inter_range_inr
principal_empty
map_inr_inf_map_inl πŸ“–mathematicalβ€”Filter
instInf
map
Bot.bot
instBot
β€”inf_comm
map_inl_inf_map_inr
map_le_iff_le_comap πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
comap
β€”mem_of_superset
Set.Subset.rfl
map_le_map_iff πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
β€”map_le_iff_le_comap
comap_map
map_le_map_iff_of_injOn πŸ“–mathematicalSet
Filter
instMembership
Set.InjOn
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
map
β€”mp_mem
mem_of_superset
image_mem_map
inter_mem
map_mono
map_map πŸ“–mathematicalβ€”mapβ€”map_compose
map_mono πŸ“–mathematicalβ€”Monotone
Filter
PartialOrder.toPreorder
instPartialOrder
map
β€”GaloisConnection.monotone_l
gc_map_comap
map_neBot πŸ“–mathematicalβ€”NeBot
map
β€”NeBot.map
map_neBot_iff πŸ“–mathematicalβ€”NeBot
map
β€”β€”
map_principal πŸ“–mathematicalβ€”map
principal
Set.image
β€”ext
Set.image_subset_iff
map_pure πŸ“–mathematicalβ€”map
Filter
instPure
β€”β€”
map_sumElim_eq πŸ“–mathematicalβ€”map
Filter
instSup
comap
β€”map_comap_inl_sup_map_comap_inr
map_sup
map_map
comap_sup
GaloisConnection.u_l_u_eq_u
gc_map_comap
comap_inl_map_inr
sup_of_le_left
comap_inr_map_inl
sup_of_le_right
map_sup πŸ“–mathematicalβ€”map
Filter
instSup
β€”GaloisConnection.l_sup
gc_map_comap
map_surjOn_Iic_iff_le_map πŸ“–mathematicalβ€”Set.SurjOn
Filter
map
Set.Iic
PartialOrder.toPreorder
instPartialOrder
Preorder.toLE
β€”Set.self_mem_Iic
map_mono
push_pull
LE.le.trans
Set.RightInvOn.surjOn
Set.mem_Iic
inf_le_left
map_surjOn_Iic_iff_surjOn πŸ“–mathematicalβ€”Set.SurjOn
Filter
map
Set.Iic
PartialOrder.toPreorder
instPartialOrder
principal
β€”map_surjOn_Iic_iff_le_map
map_principal
principal_mono
Set.SurjOn.eq_1
map_swap4_eq_comap πŸ“–mathematicalβ€”map
comap
β€”map_eq_comap_of_inverse
map_swap_eq_comap_swap πŸ“–mathematicalβ€”map
comap
β€”map_eq_comap_of_inverse
map_top πŸ“–mathematicalβ€”map
Top.top
Filter
instTop
principal
Set.range
β€”principal_univ
map_principal
Set.image_univ
mem_bind πŸ“–mathematicalβ€”Set
Filter
instMembership
bind
β€”exists_mem_subset_iff
mem_bind' πŸ“–mathematicalβ€”Set
Filter
instMembership
bind
setOf
β€”β€”
mem_comap πŸ“–mathematicalβ€”Set
Filter
instMembership
comap
Set.instHasSubset
Set.preimage
β€”β€”
mem_comap' πŸ“–mathematicalβ€”Set
Filter
instMembership
comap
setOf
β€”mem_of_superset
Set.mem_preimage
mem_comap'' πŸ“–mathematicalβ€”Set
Filter
instMembership
comap
Set.kernImage
β€”mem_comap'
mem_comap_iff πŸ“–mathematicalSet
Filter
instMembership
Set.range
comap
Set.image
β€”image_mem_map_iff
map_comap_of_mem
mem_comap_iff_compl πŸ“–mathematicalβ€”Set
Filter
instMembership
comap
Compl.compl
Set.instCompl
Set.image
β€”Set.kernImage_eq_compl
mem_comap_prodMk πŸ“–mathematicalβ€”Set
Filter
instMembership
comap
setOf
β€”forall_swap
mem_kernMap πŸ“–mathematicalβ€”Set
Filter
instMembership
kernMap
Set.kernImage
β€”β€”
mem_kernMap_iff_compl πŸ“–mathematicalβ€”Set
Filter
instMembership
kernMap
Compl.compl
Set.instCompl
Set.image
β€”mem_kernMap
Function.Surjective.exists
compl_surjective
Set.kernImage_compl
compl_eq_comm
mem_map πŸ“–mathematicalβ€”Set
Filter
instMembership
map
Set.preimage
β€”β€”
mem_map' πŸ“–mathematicalβ€”Set
Filter
instMembership
map
setOf
Set.instMembership
β€”β€”
mem_map_iff_exists_image πŸ“–mathematicalβ€”Set
Filter
instMembership
map
Set.instHasSubset
Set.image
β€”Set.image_preimage_subset
mem_of_superset
image_mem_map
mem_map_seq_iff πŸ“–mathematicalβ€”Set
Filter
instMembership
seq
map
Set.instMembership
β€”image_mem_map
mem_seq_def πŸ“–mathematicalβ€”Set
Filter
instMembership
seq
Set.instMembership
β€”β€”
mem_seq_iff πŸ“–mathematicalβ€”Set
Filter
instMembership
seq
Set.instHasSubset
Set.seq
β€”β€”
neBot_inf_comap_iff_map πŸ“–mathematicalβ€”NeBot
Filter
instInf
comap
map
β€”map_neBot_iff
push_pull
neBot_inf_comap_iff_map' πŸ“–mathematicalβ€”NeBot
Filter
instInf
comap
map
β€”map_neBot_iff
push_pull'
neBot_of_comap πŸ“–mathematicalβ€”NeBotβ€”neBot_iff
Mathlib.Tactic.Contrapose.contraposeβ‚„
comap_bot
preimage_mem_comap πŸ“–mathematicalSet
Filter
instMembership
comap
Set.preimage
β€”Set.Subset.rfl
principal_bind πŸ“–mathematicalβ€”bind
principal
iSup
Filter
instSupSet
Set
Set.instMembership
β€”map_principal
sSup_image
principal_eq_map_coe_top πŸ“–mathematicalβ€”principal
map
Set
Set.instMembership
Top.top
Filter
instTop
β€”map_top
Subtype.range_coe_subtype
principal_singleton πŸ“–mathematicalβ€”principal
Set
Set.instSingletonSet
Filter
instPure
β€”ext
principal_subtype πŸ“–mathematicalβ€”principal
Set.Elem
comap
Set
Set.instMembership
Set.image
β€”comap_principal
Set.preimage_image_eq
Subtype.coe_injective
prod_map_seq_comm πŸ“–mathematicalβ€”seq
map
β€”le_antisymm
le_seq
mem_map_iff_exists_image
mem_of_superset
Set.prod_image_seq_comm
seq_mem_seq
image_mem_map
Set.seq_mono
Set.Subset.rfl
pure_bind πŸ“–mathematicalβ€”bind
Filter
instPure
β€”β€”
pure_injective πŸ“–mathematicalβ€”Filter
instPure
β€”ext_iff
pure_le_principal πŸ“–mathematicalβ€”Filter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
instPure
principal
Set
Set.instMembership
β€”β€”
pure_neBot πŸ“–mathematicalβ€”NeBot
Filter
instPure
β€”empty_mem_iff_bot
Set.notMem_empty
pure_seq_eq_map πŸ“–mathematicalβ€”seq
Filter
instPure
map
β€”le_antisymm
le_map
Set.singleton_seq
seq_mem_seq
singleton_mem_pure
le_seq
sets_of_superset
image_mem_map
pure_sets πŸ“–mathematicalβ€”sets
Filter
instPure
setOf
Set
Set.instMembership
β€”β€”
push_pull πŸ“–mathematicalβ€”map
Filter
instInf
comap
β€”le_antisymm
map_inf_le
inf_le_inf_left
map_comap_le
mem_inf_of_inter
image_mem_map
Set.image_inter_preimage
Set.image_mono
Set.inter_subset_inter
subset_refl
Set.instReflSubset
Set.image_preimage_subset
push_pull' πŸ“–mathematicalβ€”map
Filter
instInf
comap
β€”inf_comm
push_pull
range_mem_map πŸ“–mathematicalβ€”Set
Filter
instMembership
map
Set.range
β€”Set.image_univ
image_mem_map
univ_mem
sInter_comap_sets πŸ“–mathematicalβ€”Set.sInter
sets
comap
Set.iInter
Set
Filter
instMembership
Set.preimage
β€”Set.ext
seq_assoc πŸ“–mathematicalβ€”seq
map
β€”le_antisymm
le_seq
mem_seq_iff
mem_map_iff_exists_image
mem_of_superset
Set.seq_seq
seq_mem_seq
Set.seq_mono
HasSubset.Subset.trans
Set.instIsTransSubset
Set.Subset.rfl
image_mem_map
seq_eq_filter_seq πŸ“–mathematicalβ€”Filter
instAlternative
seq
β€”β€”
seq_mem_seq πŸ“–mathematicalSet
Filter
instMembership
seq
Set.seq
β€”β€”
seq_mono πŸ“–mathematicalFilter
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
seqβ€”le_seq
seq_mem_seq
seq_pure πŸ“–mathematicalβ€”seq
Filter
instPure
map
β€”le_antisymm
le_map
Set.seq_singleton
seq_mem_seq
singleton_mem_pure
le_seq
sets_of_superset
image_mem_map
singleton_mem_pure πŸ“–mathematicalβ€”Set
Filter
instMembership
instPure
Set.instSingletonSet
β€”Set.mem_singleton
subtype_coe_map_comap πŸ“–mathematicalβ€”map
Set
Set.instMembership
comap
Filter
instInf
principal
β€”map_comap
Subtype.range_coe
sup_bind πŸ“–mathematicalβ€”bind
Filter
instSup
β€”β€”

Filter.Eventually

Theorems

NameKindAssumesProvesValidatesDepends On
comap πŸ“–mathematicalFilter.EventuallyFilter.comapβ€”Filter.preimage_mem_comap

Filter.NeBot

Theorems

NameKindAssumesProvesValidatesDepends On
comap_of_image_mem πŸ“–mathematicalSet
Filter
Filter.instMembership
Set.image
Filter.NeBot
Filter.comap
β€”comap_of_range_mem
Filter.mem_of_superset
Set.image_subset_range
comap_of_range_mem πŸ“–mathematicalSet
Filter
Filter.instMembership
Set.range
Filter.NeBot
Filter.comap
β€”Filter.comap_neBot_iff_frequently
Filter.Eventually.frequently
comap_of_surj πŸ“–mathematicalβ€”Filter.NeBot
Filter.comap
β€”comap_of_range_mem
Filter.univ_mem'
map πŸ“–mathematicalβ€”Filter.NeBot
Filter.map
β€”Filter.map_neBot_iff
of_map πŸ“–mathematicalβ€”Filter.NeBotβ€”Filter.map_neBot_iff

Function.Commute

Theorems

NameKindAssumesProvesValidatesDepends On
filter_comap πŸ“–mathematicalFunction.CommuteFilter
Filter.comap
β€”Function.Semiconj.filter_comap
semiconj
filter_map πŸ“–mathematicalFunction.CommuteFilter
Filter.map
β€”Function.Semiconj.filter_map
semiconj

Function.LeftInverse

Theorems

NameKindAssumesProvesValidatesDepends On
filter_comap πŸ“–mathematicalβ€”Filter
Filter.comap
β€”Filter.comap_comap
comp_eq_id
Filter.comap_id
filter_map πŸ“–mathematicalβ€”Filter
Filter.map
β€”Filter.map_map
comp_eq_id
Filter.map_id

Function.RightInverse

Theorems

NameKindAssumesProvesValidatesDepends On
filter_comap πŸ“–mathematicalβ€”Filter
Filter.comap
β€”Function.LeftInverse.filter_comap
filter_map πŸ“–mathematicalβ€”Filter
Filter.map
β€”Function.LeftInverse.filter_map

Function.Semiconj

Theorems

NameKindAssumesProvesValidatesDepends On
filter_comap πŸ“–mathematicalFunction.SemiconjFilter
Filter.comap
β€”Filter.comap_comm
comp_eq
filter_map πŸ“–mathematicalFunction.SemiconjFilter
Filter.map
β€”Filter.map_comm
comp_eq

Function.Surjective

Theorems

NameKindAssumesProvesValidatesDepends On
filter_map_top πŸ“–mathematicalβ€”Filter.map
Top.top
Filter
Filter.instTop
β€”Filter.comap_top
Filter.map_comap_of_surjective

Mathlib.Tactic.Linarith

Definitions

NameCategoryTheorems
Map πŸ“–CompOpβ€”

Relation

Definitions

NameCategoryTheorems
Map πŸ“–MathDef
19 mathmath: onFun_map_onFun_iff_onFun, Sym2.fromRel_relationMap, map_apply_apply, RingCon.coe_ofMatrix_eq_relationMap, map_onFun_eq_of_surjective, map_onFun_map_eq_map, map_map, onFun_map_eq_of_injective, SimpleGraph.Subgraph.map_adj, map_apply, map_onFun_le, map_id_id, onFun_map_onFun_eq_onFun, map_symmetric, map_equivalence, le_onFun_map, map_transitive, map_reflexive, RingCon.mapGen_eq_map_of_surjective

Set.InjOn

Theorems

NameKindAssumesProvesValidatesDepends On
filter_map_Iic πŸ“–mathematicalSet.InjOnFilter
Filter.map
Set.Iic
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
β€”Filter.filter_injOn_Iic_iff_injOn

Set.LeftInvOn

Theorems

NameKindAssumesProvesValidatesDepends On
filter_map_Iic πŸ“–mathematicalSet.LeftInvOnFilter
Filter.map
Set.Iic
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
β€”Filter.map_map
Filter.map_congr
Filter.EventuallyEq.filter_mono
Filter.map_id

Set.RightInvOn

Theorems

NameKindAssumesProvesValidatesDepends On
filter_map_Iic πŸ“–mathematicalSet.RightInvOnFilter
Filter.map
Set.Iic
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
β€”Set.LeftInvOn.filter_map_Iic

Set.SurjOn

Theorems

NameKindAssumesProvesValidatesDepends On
filter_map_Iic πŸ“–mathematicalSet.SurjOnFilter
Filter.map
Set.Iic
PartialOrder.toPreorder
Filter.instPartialOrder
Filter.principal
β€”Filter.map_surjOn_Iic_iff_surjOn

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