Nuprl Lemma : sv-bag-only-filter
∀[A:Type]. ∀[b:bag(A)]. ∀[p:{x:A| x ↓∈ b} ⟶ 𝔹].
∀x:A. (sv-bag-only([x∈b|p[x]]) = x ∈ A) supposing ((↑p[x]) and x ↓∈ b and (∀y:A. (y ↓∈ b ⇒ (↑p[y]) ⇒ (y = x ∈ A))))
Proof
Definitions occuring in Statement :
sv-bag-only: sv-bag-only(b),
bag-member: x ↓∈ bs,
bag-filter: [x∈b|p[x]],
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B
Lemmas referenced :
bag-filter-wf2,
subtype_rel_bag,
bag-member_wf,
assert_wf,
single-valued-bag-filter,
bag-member-size,
bag-member-filter2,
bag-member-sv-bag-only,
sv-bag-only_wf,
all_wf,
equal_wf,
bool_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
applyEquality,
setEquality,
cumulativity,
functionExtensionality,
setElimination,
rename,
dependent_set_memberEquality,
independent_isectElimination,
lambdaEquality,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
isect_memberEquality,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[b:bag(A)]. \mforall{}[p:\{x:A| x \mdownarrow{}\mmember{} b\} {}\mrightarrow{} \mBbbB{}].
\mforall{}x:A
(sv-bag-only([x\mmember{}b|p[x]]) = x) supposing
((\muparrow{}p[x]) and
x \mdownarrow{}\mmember{} b and
(\mforall{}y:A. (y \mdownarrow{}\mmember{} b {}\mRightarrow{} (\muparrow{}p[y]) {}\mRightarrow{} (y = x))))
Date html generated:
2017_10_01-AM-08_57_48
Last ObjectModification:
2017_07_26-PM-04_39_57
Theory : bags
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