Nuprl Lemma : l_exists_map
∀[A,B:Type]. ∀f:A ⟶ B. ∀L:A List. ∀[P:B ⟶ ℙ]. ((∃x∈map(f;L). P[x]) ⇐⇒ (∃x∈L. P[f x]))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x]
Lemmas referenced :
subtype_rel_self,
iff_weakening_equal,
l_member_wf,
member_map,
map_wf,
l_exists_iff,
l_exists_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
Error :dependent_pairFormation_alt,
hypothesisEquality,
hypothesis,
applyEquality,
equalitySymmetry,
sqequalRule,
instantiate,
introduction,
extract_by_obid,
isectElimination,
universeEquality,
equalityTransitivity,
independent_isectElimination,
independent_functionElimination,
Error :productIsType,
Error :universeIsType,
because_Cache,
Error :equalityIsType1,
Error :inhabitedIsType,
dependent_functionElimination,
promote_hyp,
Error :lambdaEquality_alt,
setElimination,
rename,
functionExtensionality,
cumulativity,
Error :setIsType,
Error :functionIsType
Latex:
\mforall{}[A,B:Type]. \mforall{}f:A {}\mrightarrow{} B. \mforall{}L:A List. \mforall{}[P:B {}\mrightarrow{} \mBbbP{}]. ((\mexists{}x\mmember{}map(f;L). P[x]) \mLeftarrow{}{}\mRightarrow{} (\mexists{}x\mmember{}L. P[f x]))
Date html generated:
2019_06_20-PM-00_41_41
Last ObjectModification:
2018_10_01-PM-08_40_24
Theory : list_0
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