Nuprl Lemma : fix_wf_coSet_system
∀[I:𝕌']. ∀[G:⋂C:{C:𝕌'| ((T:Type × (T ⟶ C)) ⊆r C) ∧ (coSet{i:l} ⊆r C)} . ((I ⟶ C) ⟶ I ⟶ (T:Type × (T ⟶ C)))].
(fix(G) ∈ I ⟶ coSet{i:l})
Proof
Definitions occuring in Statement :
coSet: coSet{i:l},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
fix: fix(F),
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
cand: A c∧ B,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
type-monotone: Monotone(T.F[T]),
uimplies: b supposing a,
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
corec-subtype-coSet,
subtype_rel_dep_function,
set_wf,
subtype_rel_transitivity,
coSet-subtype-corec,
subtype_rel_sets,
subtype_rel_function,
subtype_rel_product,
fix_wf_corec_system,
coSet_wf,
subtype_rel_wf,
corec_wf
Rules used in proof :
independent_pairFormation,
productElimination,
applyEquality,
lambdaFormation,
independent_isectElimination,
because_Cache,
isect_memberEquality,
rename,
setElimination,
setEquality,
isectEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
hypothesis,
hypothesisEquality,
cumulativity,
functionEquality,
universeEquality,
productEquality,
lambdaEquality,
sqequalRule,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
instantiate,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[I:\mBbbU{}']. \mforall{}[G:\mcap{}C:\{C:\mBbbU{}'| ((T:Type \mtimes{} (T {}\mrightarrow{} C)) \msubseteq{}r C) \mwedge{} (coSet\{i:l\} \msubseteq{}r C)\}
((I {}\mrightarrow{} C) {}\mrightarrow{} I {}\mrightarrow{} (T:Type \mtimes{} (T {}\mrightarrow{} C)))].
(fix(G) \mmember{} I {}\mrightarrow{} coSet\{i:l\})
Date html generated:
2018_07_29-AM-09_50_15
Last ObjectModification:
2018_07_11-AM-11_04_28
Theory : constructive!set!theory
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