Nuprl Lemma : member-fset-mapfilter
∀[T:Type]. ∀[eqT:EqDecider(T)]. ∀[P:T ⟶ 𝔹]. ∀[X:Type]. ∀[eqX:EqDecider(X)]. ∀[f:{x:T| ↑(P x)} ⟶ X]. ∀[s:fset(T)].
∀[x:X].
uiff(x ∈ fset-mapfilter(f;P;s);↓∃y:T. (y ∈ s ∧ (↑(P y)) ∧ (x = (f y) ∈ X)))
Proof
Definitions occuring in Statement :
fset-mapfilter: fset-mapfilter(f;P;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: 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],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
quotient: x,y:A//B[x; y],
so_apply: x[s],
so_lambda: λ2x.t[x],
fset: fset(T),
cand: A c∧ B,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
fset-member: a ∈ s,
fset-mapfilter: fset-mapfilter(f;P;s),
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
true: True,
false: False,
not: ¬A,
istype: istype(T)
Lemmas referenced :
fset-member_wf,
fset-mapfilter_wf,
fset-member_witness,
squash_wf,
assert_wf,
equal_wf,
istype-assert,
fset_wf,
deq_wf,
bool_wf,
istype-universe,
set-equal_wf,
list_wf,
equal-wf-base,
exists_wf,
set_wf,
subtype_rel_self,
l_member_wf,
subtype_rel_dep_function,
member-mapfilter,
mapfilter_wf,
assert-deq-member,
list_subtype_fset,
decidable__fset-member,
subtype_base_sq,
int_subtype_base,
set-equal-reflex
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
Error :universeIsType,
extract_by_obid,
isectElimination,
independent_functionElimination,
productEquality,
applyEquality,
Error :dependent_set_memberEquality_alt,
productElimination,
independent_pairEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
because_Cache,
Error :functionIsType,
Error :setIsType,
instantiate,
universeEquality,
pertypeElimination,
dependent_set_memberEquality,
setEquality,
functionExtensionality,
lambdaEquality,
cumulativity,
pointwiseFunctionalityForEquality,
lambdaFormation,
rename,
setElimination,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
dependent_pairFormation,
unionElimination,
intEquality,
natural_numberEquality,
voidElimination,
promote_hyp,
Error :lambdaFormation_alt,
pointwiseFunctionality,
Error :lambdaEquality_alt,
Error :productIsType,
Error :equalityIsType4,
Error :dependent_pairFormation_alt,
Error :equalityIstype
Latex:
\mforall{}[T:Type]. \mforall{}[eqT:EqDecider(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[X:Type]. \mforall{}[eqX:EqDecider(X)].
\mforall{}[f:\{x:T| \muparrow{}(P x)\} {}\mrightarrow{} X]. \mforall{}[s:fset(T)]. \mforall{}[x:X].
uiff(x \mmember{} fset-mapfilter(f;P;s);\mdownarrow{}\mexists{}y:T. (y \mmember{} s \mwedge{} (\muparrow{}(P y)) \mwedge{} (x = (f y))))
Date html generated:
2019_06_20-PM-01_58_50
Last ObjectModification:
2018_11_23-PM-02_42_36
Theory : finite!sets
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