Nuprl Lemma : relclosed-iff-funclosed
∀R:Set{i:l} ⟶ Set{i:l} ⟶ ℙ
((∀x:Set{i:l}. ∃y:Set{i:l}. ∀a:Set{i:l}. (R[x;a] ⇐⇒ (a ∈ y)))
⇒ (∃f:Set{i:l} ⟶ Set{i:l}. ∀s:Set{i:l}. (closed(x,a.R[x;a])s ⇐⇒ f-closed(s))))
Proof
Definitions occuring in Statement :
funclosed-set: f-closed(s),
relclosed-set: closed(x,a.R[x; a])s,
setmem: (x ∈ s),
Set: Set{i:l},
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
guard: {T},
subtype_rel: A ⊆r B,
relclosed-set: closed(x,a.R[x; a])s,
funclosed-set: f-closed(s),
pi1: fst(t),
and: P ∧ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
setsubset-iff,
subtype_rel_self,
setsubset_wf,
equal_wf,
setmem_wf,
exists_wf,
funclosed-set_wf,
relclosed-set_wf,
iff_wf,
Set_wf,
all_wf
Rules used in proof :
impliesLevelFunctionality,
allLevelFunctionality,
allFunctionality,
impliesFunctionality,
addLevel,
independent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
functionExtensionality,
rename,
universeEquality,
functionEquality,
cumulativity,
applyEquality,
lambdaEquality,
sqequalRule,
isectElimination,
extract_by_obid,
introduction,
instantiate,
because_Cache,
hypothesisEquality,
dependent_pairFormation,
productElimination,
sqequalHypSubstitution,
thin,
promote_hyp,
hypothesis,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}R:Set\{i:l\} {}\mrightarrow{} Set\{i:l\} {}\mrightarrow{} \mBbbP{}
((\mforall{}x:Set\{i:l\}. \mexists{}y:Set\{i:l\}. \mforall{}a:Set\{i:l\}. (R[x;a] \mLeftarrow{}{}\mRightarrow{} (a \mmember{} y)))
{}\mRightarrow{} (\mexists{}f:Set\{i:l\} {}\mrightarrow{} Set\{i:l\}. \mforall{}s:Set\{i:l\}. (closed(x,a.R[x;a])s \mLeftarrow{}{}\mRightarrow{} f-closed(s))))
Date html generated:
2018_05_29-PM-01_55_04
Last ObjectModification:
2018_05_25-AM-08_55_36
Theory : constructive!set!theory
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