Nuprl Lemma : member-map
∀[T,T':Type]. ∀f:T ⟶ T'. ∀a:T List. ∀x:T'. ((x ∈ map(f;a)) ⇐⇒ ∃y:T. ((y ∈ a) ∧ (x = (f y) ∈ T')))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
exists_wf,
and_wf,
l_member_wf,
equal_wf,
member_map,
map_wf,
iff_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
universeEquality
Latex:
\mforall{}[T,T':Type]. \mforall{}f:T {}\mrightarrow{} T'. \mforall{}a:T List. \mforall{}x:T'. ((x \mmember{} map(f;a)) \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((y \mmember{} a) \mwedge{} (x = (f y))))
Date html generated:
2016_05_14-AM-06_42_39
Last ObjectModification:
2015_12_26-PM-00_29_05
Theory : list_0
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