Nuprl Lemma : set-metric-subspace
∀[X:Type]. ∀[d:metric(X)]. ∀[P:X ⟶ ℙ]. metric-subspace(X;d;{x:X| P[x]} ) supposing ∀x,y:X. (P[x] ⇒ y ≡ x ⇒ P[y])
Proof
Definitions occuring in Statement :
metric-subspace: metric-subspace(X;d;A),
meq: x ≡ y,
metric: metric(X),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
metric-subspace: metric-subspace(X;d;A),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Lemmas referenced :
strong-subtype-set2,
meq_wf,
strong-subtype_witness,
subtype_rel_self,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
universeIsType,
hypothesis,
lambdaFormation_alt,
setElimination,
rename,
dependent_set_memberEquality_alt,
because_Cache,
setIsType,
productElimination,
independent_pairEquality,
setEquality,
instantiate,
universeEquality,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[P:X {}\mrightarrow{} \mBbbP{}]. metric-subspace(X;d;\{x:X| P[x]\} ) supposing \mforall{}x,y:X. (P[x] \000C{}\mRightarrow{} y \mequiv{} x {}\mRightarrow{} P[y])
Date html generated:
2019_10_30-AM-06_31_08
Last ObjectModification:
2019_10_02-AM-10_06_00
Theory : reals
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