Nuprl Lemma : fset-filter_wf
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[s:fset(T)]. ({x ∈ s | P[x]} ∈ fset(T))
Proof
Definitions occuring in Statement :
fset-filter: {x ∈ s | P[x]},
fset: fset(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fset: fset(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
fset-filter: {x ∈ s | P[x]},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
set-equal: set-equal(T;x;y),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
quotient-member-eq,
list_wf,
set-equal_wf,
set-equal-equiv,
filter_wf5,
l_member_wf,
equal-wf-base,
fset_wf,
bool_wf,
member_filter,
iff_wf,
and_wf,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
because_Cache,
sqequalRule,
pertypeElimination,
productElimination,
thin,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
setElimination,
rename,
setEquality,
independent_functionElimination,
productEquality,
cumulativity,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
universeEquality,
lambdaFormation,
addLevel,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)]. (\{x \mmember{} s | P[x]\} \mmember{} fset(T))
Date html generated:
2016_05_14-PM-03_39_21
Last ObjectModification:
2015_12_26-PM-06_41_53
Theory : finite!sets
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