Nuprl Lemma : fset-filter_wf2
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[s:fset(T)]. ({x ∈ s | P[x]} ∈ fset({x:T| ↑P[x]} ))
Proof
Definitions occuring in Statement :
fset-filter: {x ∈ s | P[x]},
fset: fset(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fset: fset(T),
so_apply: x[s],
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
fset-filter: {x ∈ s | P[x]},
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
set-equal: set-equal(T;x;y),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fset_wf,
assert_wf,
filter_type,
list_wf,
quotient-member-eq,
set-equal_wf,
set-equal-equiv,
filter_wf3,
member-filter,
l_member_wf,
iff_wf,
equal-wf-base,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
lemma_by_obid,
isectElimination,
thin,
setEquality,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
lambdaFormation,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
addLevel,
independent_pairFormation,
impliesFunctionality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productEquality,
cumulativity,
axiomEquality,
isect_memberEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)]. (\{x \mmember{} s | P[x]\} \mmember{} fset(\{x:T| \muparrow{}P[x]\} ))
Date html generated:
2016_05_14-PM-03_39_23
Last ObjectModification:
2015_12_26-PM-06_41_40
Theory : finite!sets
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