Nuprl Lemma : fset-powerset_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. (fset-powerset(eq;s) ∈ {p:fset(fset(T))| ∀x:fset(T). (x ∈ p ⇐⇒ x ⊆ s)} )
Proof
Definitions occuring in Statement :
fset-powerset: fset-powerset(eq;s),
deq-fset: deq-fset(eq),
f-subset: xs ⊆ ys,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fset: fset(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
quotient: x,y:A//B[x; y],
fset-powerset: fset-powerset(eq;s),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
f-subset: xs ⊆ ys
Lemmas referenced :
fset_wf,
all_wf,
iff_wf,
fset-member_wf,
deq-fset_wf,
f-subset_wf,
list-powerset_wf,
set_wf,
list_subtype_fset,
equal_wf,
equal-wf-base,
list_wf,
set-equal_wf,
deq_wf,
fset-extensionality,
fset-member_witness,
uiff_wf,
quotient-member-eq,
set-equal-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
setEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
pertypeElimination,
productElimination,
because_Cache,
applyEquality,
independent_isectElimination,
lambdaFormation,
setElimination,
rename,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productEquality,
axiomEquality,
isect_memberEquality,
universeEquality,
independent_pairEquality,
addLevel,
independent_pairFormation,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)].
(fset-powerset(eq;s) \mmember{} \{p:fset(fset(T))| \mforall{}x:fset(T). (x \mmember{} p \mLeftarrow{}{}\mRightarrow{} x \msubseteq{} s)\} )
Date html generated:
2016_10_21-AM-10_47_09
Last ObjectModification:
2016_07_12-AM-05_52_46
Theory : finite!sets
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