Nuprl Lemma : all_fset_elim
∀s:DSet. ∀F:MSet{s} ⟶ ℙ.
((∀a:FiniteSet{s}. SqStable(F[a])) ⇒ (∀a:FiniteSet{s}. F[a] ⇐⇒ ∀a:DisList{s}. F[mk_mset(a)]))
Proof
Definitions occuring in Statement :
finite_set: FiniteSet{s},
mk_mset: mk_mset(as),
mset: MSet{s},
dislist: DisList{s},
sq_stable: SqStable(P),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
dset: DSet
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
finite_set: FiniteSet{s},
rev_implies: P ⇐ Q,
dislist: DisList{s},
sq_stable: SqStable(P),
dset: DSet,
subtype_rel: A ⊆r B,
nat: ℕ,
squash: ↓T,
mset: MSet{s},
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
mk_mset: mk_mset(as),
mset_count: x #∈ a
Lemmas referenced :
dislist_wf,
all_wf,
finite_set_wf,
mset_wf,
mk_mset_wf,
sq_stable_wf,
dset_wf,
mk_mset_wf2,
sq_stable__all,
set_car_wf,
le_wf,
mset_count_wf,
nat_wf,
squash_wf,
sq_stable__squash,
list_wf,
subtype_quotient,
permr_wf,
permr_equiv_rel,
equal_wf,
equal-wf-base,
count_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
natural_numberEquality,
because_Cache,
imageElimination,
imageMemberEquality,
baseClosed,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productEquality,
dependent_set_memberEquality
Latex:
\mforall{}s:DSet. \mforall{}F:MSet\{s\} {}\mrightarrow{} \mBbbP{}.
((\mforall{}a:FiniteSet\{s\}. SqStable(F[a])) {}\mRightarrow{} (\mforall{}a:FiniteSet\{s\}. F[a] \mLeftarrow{}{}\mRightarrow{} \mforall{}a:DisList\{s\}. F[mk\_mset(a)]))
Date html generated:
2017_10_01-AM-09_59_03
Last ObjectModification:
2017_03_03-PM-01_00_18
Theory : mset
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