Nuprl Lemma : member-mapl
∀[T,T':Type]. ∀L:T List. ∀y:T'. ∀f:{x:T| (x ∈ L)} ⟶ T'. ((y ∈ mapl(f;L)) ⇐⇒ ∃a:T. ((a ∈ L) c∧ (y = (f a) ∈ T')))
Proof
Definitions occuring in Statement :
mapl: mapl(f;l),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
cand: A c∧ B,
so_apply: x[s],
implies: P ⇒ Q,
mapl: mapl(f;l),
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
subtype_rel: A ⊆r B,
guard: {T},
squash: ↓T
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
iff_wf,
mapl_wf,
exists_wf,
equal_wf,
list_wf,
map_nil_lemma,
map_cons_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
length_of_nil_lemma,
stuck-spread,
base_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
cons_wf,
cons_member,
subtype_rel_dep_function,
subtype_rel_sets,
set_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
functionEquality,
setEquality,
because_Cache,
hypothesis,
setElimination,
rename,
functionExtensionality,
applyEquality,
productEquality,
dependent_set_memberEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
universeEquality,
independent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
baseClosed,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
inlFormation,
inrFormation,
unionElimination,
hyp_replacement,
applyLambdaEquality,
imageMemberEquality,
imageElimination
Latex:
\mforall{}[T,T':Type].
\mforall{}L:T List. \mforall{}y:T'. \mforall{}f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} T'. ((y \mmember{} mapl(f;L)) \mLeftarrow{}{}\mRightarrow{} \mexists{}a:T. ((a \mmember{} L) c\mwedge{} (y = (f a))))
Date html generated:
2017_04_17-AM-08_41_03
Last ObjectModification:
2017_02_27-PM-05_00_52
Theory : list_1
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