Nuprl Lemma : bexists_char
∀s:DSet. ∀as:|s| List. ∀f:|s| ⟶ 𝔹. (↑(∃bx(:|s|) ∈ as. f[x]) ⇐⇒ ∃x:|s|. ((↑(x ∈b as)) ∧ (↑f[x])))
Proof
Definitions occuring in Statement :
bexists: bexists,
mem: a ∈b as,
list: T List,
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
implies: P ⇒ Q,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
false: False,
or: P ∨ Q,
infix_ap: x f y,
guard: {T},
cand: A c∧ B
Lemmas referenced :
set_car_wf,
bool_wf,
list_wf,
dset_wf,
list_induction,
iff_wf,
assert_wf,
bexists_wf,
exists_wf,
mem_wf,
bexists_nil_lemma,
istype-void,
mem_nil_lemma,
bexists_cons_lemma,
mem_cons_lemma,
bor_wf,
or_wf,
set_eq_wf,
equal_wf,
iff_weakening_uiff,
assert_of_bor,
iff_transitivity,
assert_of_dset_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
because_Cache,
applyEquality,
productEquality,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
productIsType,
inhabitedIsType,
independent_pairFormation,
productElimination,
unionIsType,
equalityIsType1,
dependent_pairFormation_alt,
unionElimination,
inlFormation_alt,
inrFormation_alt,
promote_hyp,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality
Latex:
\mforall{}s:DSet. \mforall{}as:|s| List. \mforall{}f:|s| {}\mrightarrow{} \mBbbB{}. (\muparrow{}(\mexists{}\msubb{}x(:|s|) \mmember{} as. f[x]) \mLeftarrow{}{}\mRightarrow{} \mexists{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} as)) \mwedge{} (\muparrow{}f[x])))
Date html generated:
2019_10_16-PM-01_03_31
Last ObjectModification:
2018_10_08-AM-11_25_30
Theory : list_2
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