Nuprl Lemma : mem_map
∀s,s':DSet. ∀f:|s| ⟶ |s'|. ∀x:|s|. ∀as:|s| List. ((↑(x ∈b as)) ⇒ (↑((f x) ∈b map(f;as))))
Proof
Definitions occuring in Statement :
mem: a ∈b as,
map: map(f;as),
list: T List,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
dset: DSet,
so_apply: x[s],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
infix_ap: x f y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
guard: {T},
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
list_induction,
assert_wf,
mem_wf,
map_wf,
set_car_wf,
mem_nil_lemma,
istype-void,
map_nil_lemma,
mem_cons_lemma,
map_cons_lemma,
iff_transitivity,
bor_wf,
set_eq_wf,
or_wf,
equal_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_dset_eq,
list_wf,
dset_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality_alt,
functionEquality,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
applyEquality,
setElimination,
rename,
universeIsType,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
unionElimination,
inlFormation_alt,
productElimination,
inrFormation_alt,
equalityIsType1,
unionIsType,
functionIsType,
inhabitedIsType,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination
Latex:
\mforall{}s,s':DSet. \mforall{}f:|s| {}\mrightarrow{} |s'|. \mforall{}x:|s|. \mforall{}as:|s| List. ((\muparrow{}(x \mmember{}\msubb{} as)) {}\mRightarrow{} (\muparrow{}((f x) \mmember{}\msubb{} map(f;as))))
Date html generated:
2019_10_16-PM-01_04_11
Last ObjectModification:
2018_10_08-AM-11_03_18
Theory : list_2
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