Nuprl Lemma : fset-filter-subset2
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. ∀[P:{x:T| x ∈ s} ⟶ 𝔹]. {x ∈ s | P[x]} ⊆ s
Proof
Definitions occuring in Statement :
fset-filter: {x ∈ s | P[x]},
f-subset: xs ⊆ ys,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
bool: 𝔹,
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],
f-subset: xs ⊆ ys,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
member-fset-filter2,
fset-member_wf,
fset-member_witness,
istype-universe,
bool_wf,
fset_wf,
deq_wf,
fset-filter_wf,
fset-subtype,
fset-subtype2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
hypothesis,
Error :inhabitedIsType,
setElimination,
rename,
applyEquality,
Error :dependent_set_memberEquality_alt,
Error :universeIsType,
dependent_functionElimination,
Error :setIsType,
productElimination,
independent_isectElimination,
Error :equalityIsType1,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
Error :functionIsType,
universeEquality,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. \mforall{}[P:\{x:T| x \mmember{} s\} {}\mrightarrow{} \mBbbB{}]. \{x \mmember{} s | P[x]\} \msubseteq{} s
Date html generated:
2019_06_20-PM-01_58_58
Last ObjectModification:
2018_10_06-PM-11_55_33
Theory : finite!sets
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