Nuprl Lemma : fset-mapfilter_wf
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[X:Type]. ∀[f:{x:T| ↑(P x)} ⟶ X]. ∀[s:fset(T)]. (fset-mapfilter(f;P;s) ∈ fset(X))
Proof
Definitions occuring in Statement :
fset-mapfilter: fset-mapfilter(f;P;s),
fset: fset(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fset: fset(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
fset-mapfilter: fset-mapfilter(f;P;s),
prop: ℙ,
implies: P ⇒ Q,
set-equal: set-equal(T;x;y),
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
fset_wf,
quotient-member-eq,
list_wf,
set-equal_wf,
set-equal-equiv,
mapfilter_wf,
assert_wf,
member-mapfilter,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
all_wf,
iff_wf,
exists_wf,
equal_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
functionExtensionality,
applyEquality,
setEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
addLevel,
allFunctionality,
independent_pairFormation,
impliesFunctionality,
setElimination,
rename,
dependent_set_memberEquality,
productEquality,
axiomEquality,
isect_memberEquality,
functionEquality,
universeEquality,
existsFunctionality,
andLevelFunctionality,
existsLevelFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[X:Type]. \mforall{}[f:\{x:T| \muparrow{}(P x)\} {}\mrightarrow{} X]. \mforall{}[s:fset(T)].
(fset-mapfilter(f;P;s) \mmember{} fset(X))
Date html generated:
2017_04_17-AM-09_19_13
Last ObjectModification:
2017_02_27-PM-05_22_47
Theory : finite!sets
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