Nuprl Lemma : bag-member-filter2
∀[T:Type]. ∀[x:T]. ∀[bs:bag(T)]. ∀[P:{x:T| x ↓∈ bs}  ⟶ 𝔹].  uiff(x ↓∈ [x∈bs|P[x]];x ↓∈ bs ∧ (↑P[x]))
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]
, 
and: P ∧ Q
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
uiff: uiff(P;Q)
, 
bag-member: x ↓∈ bs
, 
squash: ↓T
, 
implies: P 
⇒ Q
Lemmas referenced : 
assert_witness, 
bag_wf, 
bool_wf, 
bag-member-filter-implies1, 
assert_wf, 
subtype_rel_bag, 
bag-member_wf, 
bag-filter-wf2, 
bag-member-filter-implies2
Rules used in proof : 
cut, 
lemma_by_obid, 
sqequalHypSubstitution, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isectElimination, 
thin, 
because_Cache, 
hypothesisEquality, 
independent_isectElimination, 
hypothesis, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
setEquality, 
dependent_set_memberEquality, 
cumulativity, 
equalityTransitivity, 
equalitySymmetry, 
setElimination, 
rename, 
productElimination, 
independent_pairFormation, 
productEquality, 
functionEquality, 
universeEquality, 
isect_memberFormation, 
introduction, 
independent_pairEquality, 
imageElimination, 
imageMemberEquality, 
baseClosed, 
independent_functionElimination, 
isect_memberEquality
Latex:
\mforall{}[T:Type].  \mforall{}[x:T].  \mforall{}[bs:bag(T)].  \mforall{}[P:\{x:T|  x  \mdownarrow{}\mmember{}  bs\}    {}\mrightarrow{}  \mBbbB{}].    uiff(x  \mdownarrow{}\mmember{}  [x\mmember{}bs|P[x]];x  \mdownarrow{}\mmember{}  bs  \mwedge{}  (\muparrow{}P[x])\000C)
Date html generated:
2016_05_15-PM-02_47_51
Last ObjectModification:
2016_01_16-AM-08_43_19
Theory : bags
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