Nuprl Lemma : member_map2
∀[T,T':Type]. ∀a:T List. ∀x:T'. ∀f:{x:T| (x ∈ a)} ⟶ 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,
set: {x:A| B[x]} ,
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],
member: t ∈ T,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
l_member_wf,
list_wf,
member_map,
list-subtype,
l_member-settype,
equal_wf,
map_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
functionEquality,
setEquality,
cumulativity,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
universeEquality,
independent_pairFormation,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
dependent_pairFormation,
setElimination,
rename,
sqequalRule,
lambdaEquality,
because_Cache,
productEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality
Latex:
\mforall{}[T,T':Type].
\mforall{}a:T List. \mforall{}x:T'. \mforall{}f:\{x:T| (x \mmember{} a)\} {}\mrightarrow{} T'. ((x \mmember{} map(f;a)) \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((y \mmember{} a) \mwedge{} (x = (f y))))
Date html generated:
2017_04_17-AM-08_49_49
Last ObjectModification:
2017_02_27-PM-05_06_40
Theory : list_1
Home
Index