Nuprl Lemma : member_map
∀[T,T':Type]. ∀a:T List. ∀x:T'. ∀f:T ⟶ T'. ((x ∈ map(f;a)) ⇐⇒ ∃y:T. ((y ∈ a) ∧ (x = (f y) ∈ T')))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
cand: A c∧ B,
nat: ℕ,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
true: True,
guard: {T}
Lemmas referenced :
exists_wf,
nat_wf,
less_than_wf,
length_wf,
map_wf,
equal_wf,
select_wf,
sq_stable__le,
map-length,
select-map,
subtype_rel_list,
top_wf,
lelt_wf,
squash_wf,
true_wf,
iff_weakening_equal,
list_wf,
map_length,
map_select
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
lambdaEquality,
productEquality,
setElimination,
rename,
because_Cache,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
independent_isectElimination,
natural_numberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
dependent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionEquality,
intEquality
Latex:
\mforall{}[T,T':Type]. \mforall{}a:T List. \mforall{}x:T'. \mforall{}f:T {}\mrightarrow{} T'. ((x \mmember{} map(f;a)) \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((y \mmember{} a) \mwedge{} (x = (f y))))
Date html generated:
2017_04_14-AM-08_41_24
Last ObjectModification:
2017_02_27-PM-03_31_33
Theory : list_0
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