Nuprl Lemma : eclass-cond-classrel
∀[Info,B:Type]. ∀[X:EClass(B ─→ B)]. ∀[Y:EClass(B)]. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:B].
uiff(v ∈ eclass-cond(X;Y)(
e);↓if e ∈b X then ∃f:B ─→ B. ∃b:B. (f ∈ X(e) ∧ b ∈ Y(e) ∧ (v = (f b) ∈ B)) else v ∈ Y(e) fi )
Proof
Definitions occuring in Statement :
eclass-cond: eclass-cond(X;Y),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
classrel_wf,
eclass-cond_wf,
squash_wf,
member-eclass_wf,
bool_wf,
eqtt_to_assert,
exists_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
eclass_wf,
bool_cases,
assert_of_bnot,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eclass3-classrel,
sq_stable__classrel
\mforall{}[Info,B:Type]. \mforall{}[X:EClass(B {}\mrightarrow{} B)]. \mforall{}[Y:EClass(B)]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[v:B].
uiff(v \mmember{} eclass-cond(X;Y)(e);\mdownarrow{}if e \mmember{}\msubb{} X
then \mexists{}f:B {}\mrightarrow{} B. \mexists{}b:B. (f \mmember{} X(e) \mwedge{} b \mmember{} Y(e) \mwedge{} (v = (f b)))
else v \mmember{} Y(e)
fi )
Date html generated:
2015_07_17-PM-00_38_19
Last ObjectModification:
2015_01_27-PM-11_14_41
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