Nuprl Lemma : rel_finite-restrict
∀[E:Type]. ∀P:E ⟶ 𝔹. ∀[R:E ⟶ E ⟶ ℙ]. (rel_finite(E;R) ⇒ rel_finite({e:E| ↑(P e)} ;R))
Proof
Definitions occuring in Statement :
rel_finite: rel_finite(T;R),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
rel_finite: rel_finite(T;R),
member: t ∈ T,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
prop: ℙ,
infix_ap: x f y
Lemmas referenced :
filter_type,
l_member_set2,
assert_wf,
member_filter,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
set_wf,
all_wf,
l_member_wf,
rel_finite_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
productElimination,
dependent_pairFormation,
cut,
lemma_by_obid,
isectElimination,
cumulativity,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
independent_functionElimination,
independent_pairFormation,
addLevel,
independent_isectElimination,
levelHypothesis,
instantiate,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
setEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[E:Type]. \mforall{}P:E {}\mrightarrow{} \mBbbB{}. \mforall{}[R:E {}\mrightarrow{} E {}\mrightarrow{} \mBbbP{}]. (rel\_finite(E;R) {}\mRightarrow{} rel\_finite(\{e:E| \muparrow{}(P e)\} ;R))
Date html generated:
2016_05_14-PM-03_51_47
Last ObjectModification:
2015_12_26-PM-06_57_27
Theory : relations2
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