Nuprl Lemma : itersetfun-subset-fixpoint
∀G:Set{i:l} ⟶ Set{i:l}
((∀a,b:Set{i:l}. ((a ⊆ b) ⇒ (G[a] ⊆ G[b])))
⇒ (∀X:Set{i:l}. ((G[X] ⊆ X) ⇒ (∀a:Set{i:l}. (itersetfun(x.G[x];a) ⊆ X)))))
Proof
Definitions occuring in Statement :
itersetfun: itersetfun(s.G[s];a),
setsubset: (a ⊆ b),
Set: Set{i:l},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
guard: {T},
top: Top,
set-function: set-function{i:l}(s; x.f[x]),
exists: ∃x:A. B[x],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
itersetfun: itersetfun(s.G[s];a),
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
setsubset-iff,
setmem_functionality,
coSet-seteq-Set,
setmem-mk-set-sq,
seteq_wf,
seteq_weakening,
itersetfun_functionality,
seteq_functionality,
setmem-setunionfun,
coSet-mem-Set-implies-Set,
coSet_wf,
setunionfun_wf,
setsubset-iff2,
setmem_wf,
setunionfun_wf2,
all_wf,
mk-set_wf,
setsubset_transitivity,
set-subtype-coSet,
Set_wf,
itersetfun_wf,
setsubset_wf,
set-induction
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_pairFormation,
independent_isectElimination,
functionExtensionality,
productElimination,
setEquality,
rename,
setElimination,
instantiate,
universeEquality,
functionEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
hypothesis,
hypothesisEquality,
applyEquality,
cumulativity,
lambdaEquality,
sqequalRule,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}G:Set\{i:l\} {}\mrightarrow{} Set\{i:l\}
((\mforall{}a,b:Set\{i:l\}. ((a \msubseteq{} b) {}\mRightarrow{} (G[a] \msubseteq{} G[b])))
{}\mRightarrow{} (\mforall{}X:Set\{i:l\}. ((G[X] \msubseteq{} X) {}\mRightarrow{} (\mforall{}a:Set\{i:l\}. (itersetfun(x.G[x];a) \msubseteq{} X)))))
Date html generated:
2018_07_29-AM-10_06_03
Last ObjectModification:
2018_07_18-PM-10_13_53
Theory : constructive!set!theory
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