Nuprl Lemma : single-valued-bag-filter
∀[A:Type]. ∀[b:bag(A)]. ∀[p:{x:A| x ↓∈ b} ⟶ 𝔹].
single-valued-bag([x∈b|p[x]];A) supposing ∀x,y:A. (x ↓∈ b ⇒ y ↓∈ b ⇒ (↑p[x]) ⇒ (↑p[y]) ⇒ (x = y ∈ A))
Proof
Definitions occuring in Statement :
single-valued-bag: single-valued-bag(b;T),
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 :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
guard: {T},
single-valued-bag: single-valued-bag(b;T)
Lemmas referenced :
bag-member-filter2,
bag-member_wf,
bag-filter-wf2,
subtype_rel_bag,
assert_wf,
all_wf,
equal_wf,
bool_wf,
bag_wf
Rules used in proof :
cut,
hypothesis,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
cumulativity,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
isect_memberFormation,
lambdaFormation,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[b:bag(A)]. \mforall{}[p:\{x:A| x \mdownarrow{}\mmember{} b\} {}\mrightarrow{} \mBbbB{}].
single-valued-bag([x\mmember{}b|p[x]];A)
supposing \mforall{}x,y:A. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} y \mdownarrow{}\mmember{} b {}\mRightarrow{} (\muparrow{}p[x]) {}\mRightarrow{} (\muparrow{}p[y]) {}\mRightarrow{} (x = y))
Date html generated:
2017_10_01-AM-08_57_37
Last ObjectModification:
2017_07_26-PM-04_39_42
Theory : bags
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