Nuprl Lemma : detach_fun_properties
∀[T:Type]. ∀[A:T ⟶ ℙ]. ((∀x:T. SqStable(A x)) ⇒ (∀d:detach_fun(T;A). {∀x:T. (A x ⇐⇒ ↑(d x))}))
Proof
Definitions occuring in Statement :
detach_fun: detach_fun(T;A),
assert: ↑b,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T},
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
detach_fun: detach_fun(T;A),
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
decidable__assert,
sq_stable_from_decidable,
sq_stable__iff,
sq_stable__all,
assert_wf,
iff_wf,
sq_stable_wf,
all_wf,
detach_fun_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
setElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
introduction,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[A:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:T. SqStable(A x)) {}\mRightarrow{} (\mforall{}d:detach\_fun(T;A). \{\mforall{}x:T. (A x \mLeftarrow{}{}\mRightarrow{} \muparrow{}(d x))\}))
Date html generated:
2016_05_15-PM-00_00_24
Last ObjectModification:
2016_01_15-AM-07_08_15
Theory : gen_algebra_1
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