### Nuprl Lemma : member_map

`∀[T,T':Type].  ∀a:T List. ∀x:T'. ∀f:T ⟶ 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 :  l_member: `(x ∈ l)` 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]` cand: `A c∧ B` nat: `ℕ` uimplies: `b supposing a` sq_stable: `SqStable(P)` squash: `↓T` so_apply: `x[s]` rev_implies: `P `` Q` exists: `∃x:A. B[x]` top: `Top` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` true: `True` guard: `{T}`
Lemmas referenced :  exists_wf nat_wf less_than_wf length_wf map_wf equal_wf select_wf sq_stable__le map-length select-map subtype_rel_list top_wf lelt_wf squash_wf true_wf iff_weakening_equal list_wf map_length map_select
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis lambdaEquality productEquality setElimination rename because_Cache cumulativity hypothesisEquality functionExtensionality applyEquality independent_isectElimination natural_numberEquality independent_functionElimination imageMemberEquality baseClosed imageElimination productElimination dependent_pairFormation isect_memberEquality voidElimination voidEquality dependent_set_memberEquality equalityTransitivity equalitySymmetry universeEquality functionEquality intEquality

Latex:
\mforall{}[T,T':Type].    \mforall{}a:T  List.  \mforall{}x:T'.  \mforall{}f:T  {}\mrightarrow{}  T'.    ((x  \mmember{}  map(f;a))  \mLeftarrow{}{}\mRightarrow{}  \mexists{}y:T.  ((y  \mmember{}  a)  \mwedge{}  (x  =  (f  y))))

Date html generated: 2017_04_14-AM-08_41_24
Last ObjectModification: 2017_02_27-PM-03_31_33

Theory : list_0

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