Nuprl Lemma : closure-set-property
∀B,x:Set{i:l}. ∀Y:x1:Set{i:l} ⟶ Set{i:l}.
((∀x1,x2,a1,a2:Set{i:l}. (seteq(x1;x2) ⇒ seteq(a1;a2) ⇒ (a1 ∈ Y x1) ⇒ (a2 ∈ Y x2)))
⇒ (∀x,a:Set{i:l}. ((a ∈ Y x) ⇒ (∃b:Set{i:l}. ((b ∈ B) ∧ setimage{i:l}(x;b)))))
⇒ (∀z:Set{i:l}. ((z ∈ closure-set(B;Y;x)) ⇐⇒ ∃A:Set{i:l}. ((A ⊆ x) ∧ (z ∈ Y A)))))
Proof
Definitions occuring in Statement :
closure-set: closure-set(B;Y;x),
setimage: setimage{i:l}(x;b),
setsubset: (a ⊆ b),
Set: Set{i:l},
setmem: (x ∈ s),
seteq: seteq(s1;s2),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
subtype_rel: A ⊆r B,
cand: A c∧ B,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
set-function: set-function{i:l}(s; x.f[x]),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
setmem-setimages-2,
iff_wf,
closure-set_wf,
setmem-closure-set,
setsubset_wf,
setimages_wf2,
setimage_wf,
exists_wf,
all_wf,
Set_wf,
coSet_wf,
setimages_wf,
seteq_wf,
coSet-mem-Set-implies-Set,
setmem_wf,
seteq_inversion,
set-subtype-coSet,
seteq_weakening,
seteq-iff
Rules used in proof :
promote_hyp,
impliesFunctionality,
addLevel,
functionExtensionality,
universeEquality,
productEquality,
cumulativity,
functionEquality,
lambdaEquality,
instantiate,
dependent_pairFormation,
independent_isectElimination,
isectElimination,
independent_pairFormation,
sqequalRule,
applyEquality,
hypothesisEquality,
independent_functionElimination,
productElimination,
hypothesis,
because_Cache,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}B,x:Set\{i:l\}. \mforall{}Y:x1:Set\{i:l\} {}\mrightarrow{} Set\{i:l\}.
((\mforall{}x1,x2,a1,a2:Set\{i:l\}. (seteq(x1;x2) {}\mRightarrow{} seteq(a1;a2) {}\mRightarrow{} (a1 \mmember{} Y x1) {}\mRightarrow{} (a2 \mmember{} Y x2)))
{}\mRightarrow{} (\mforall{}x,a:Set\{i:l\}. ((a \mmember{} Y x) {}\mRightarrow{} (\mexists{}b:Set\{i:l\}. ((b \mmember{} B) \mwedge{} setimage\{i:l\}(x;b)))))
{}\mRightarrow{} (\mforall{}z:Set\{i:l\}. ((z \mmember{} closure-set(B;Y;x)) \mLeftarrow{}{}\mRightarrow{} \mexists{}A:Set\{i:l\}. ((A \msubseteq{} x) \mwedge{} (z \mmember{} Y A)))))
Date html generated:
2018_07_29-AM-10_10_11
Last ObjectModification:
2018_07_18-PM-10_33_12
Theory : constructive!set!theory
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