Nuprl Lemma : bag-member-filter-set
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[x:{x:T| ↑P[x]} ]. ∀[bs:bag(T)]. uiff(x ↓∈ [x∈bs|P[x]];x ↓∈ bs)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-filter: [x∈b|p[x]],
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
bag-member: x ↓∈ bs,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-filter: [x∈b|p[x]],
true: True
Lemmas referenced :
sq_stable__bag-member,
bag-member_wf,
assert_wf,
bag-filter_wf,
bag_wf,
set_wf,
bool_wf,
bag-member-subtype,
bag-member-filter,
filter_type,
l_member_set2,
member_filter,
equal_wf,
list-subtype-bag,
l_member_wf,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
setEquality,
cumulativity,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
lambdaEquality,
because_Cache,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
independent_isectElimination,
dependent_functionElimination,
dependent_pairFormation,
productEquality,
natural_numberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:\{x:T| \muparrow{}P[x]\} ]. \mforall{}[bs:bag(T)]. uiff(x \mdownarrow{}\mmember{} [x\mmember{}bs|P[x]];x \mdownarrow{}\mmember{} bs)
Date html generated:
2017_10_01-AM-08_54_17
Last ObjectModification:
2017_07_26-PM-04_36_01
Theory : bags
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