### Nuprl Lemma : member-mapl

`∀[T,T':Type].  ∀L:T List. ∀y:T'. ∀f:{x:T| (x ∈ L)}  ⟶ T'.  ((y ∈ mapl(f;L)) `⇐⇒` ∃a:T. ((a ∈ L) c∧ (y = (f a) ∈ T')))`

Proof

Definitions occuring in Statement :  mapl: `mapl(f;l)` l_member: `(x ∈ l)` list: `T List` uall: `∀[x:A]. B[x]` cand: `A c∧ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `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` so_lambda: `λ2x.t[x]` prop: `ℙ` cand: `A c∧ B` so_apply: `x[s]` implies: `P `` Q` mapl: `mapl(f;l)` top: `Top` iff: `P `⇐⇒` Q` and: `P ∧ Q` uimplies: `b supposing a` not: `¬A` false: `False` rev_implies: `P `` Q` exists: `∃x:A. B[x]` l_member: `(x ∈ l)` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` nat: `ℕ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` or: `P ∨ Q` subtype_rel: `A ⊆r B` guard: `{T}` squash: `↓T`
Lemmas referenced :  list_induction all_wf l_member_wf iff_wf mapl_wf exists_wf equal_wf list_wf map_nil_lemma map_cons_lemma null_nil_lemma btrue_wf member-implies-null-eq-bfalse nil_wf btrue_neq_bfalse length_of_nil_lemma stuck-spread base_wf nat_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf cons_wf cons_member subtype_rel_dep_function subtype_rel_sets set_wf and_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity functionEquality setEquality because_Cache hypothesis setElimination rename functionExtensionality applyEquality productEquality dependent_set_memberEquality independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality universeEquality independent_pairFormation independent_isectElimination equalityTransitivity equalitySymmetry productElimination baseClosed natural_numberEquality dependent_pairFormation int_eqEquality intEquality computeAll inlFormation inrFormation unionElimination hyp_replacement applyLambdaEquality imageMemberEquality imageElimination

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

Date html generated: 2017_04_17-AM-08_41_03
Last ObjectModification: 2017_02_27-PM-05_00_52

Theory : list_1

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