### Nuprl Lemma : sublist_map_inj

`∀[A,B:Type].  ∀f:A ⟶ B. ∀as,bs:A List.  (Inj(A;B;f) `` (as ⊆ bs `⇐⇒` map(f;as) ⊆ map(f;bs)))`

Proof

Definitions occuring in Statement :  sublist: `L1 ⊆ L2` map: `map(f;as)` list: `T List` inject: `Inj(A;B;f)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` sublist: `L1 ⊆ L2` exists: `∃x:A. B[x]` member: `t ∈ T` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` squash: `↓T` true: `True` guard: `{T}` rev_implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` int_seg: `{i..j-}` lelt: `i ≤ j < k` cand: `A c∧ B` less_than: `a < b` ge: `i ≥ j ` nat: `ℕ` inject: `Inj(A;B;f)`
Lemmas referenced :  subtype_rel_dep_function int_seg_wf length_wf map_wf int_seg_subtype false_wf le_wf map_length_nat iff_weakening_equal decidable__le satisfiable-full-omega-tt intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf int_seg_properties increasing_wf length_wf_nat all_wf equal_wf select_wf intformand_wf itermConstant_wf int_formula_prop_and_lemma int_term_value_constant_lemma decidable__lt intformless_wf int_formula_prop_less_lemma non_neg_length map_length lelt_wf nat_properties map-length intformeq_wf int_formula_prop_eq_lemma sublist_wf inject_wf list_wf select-map subtype_rel_list top_wf squash_wf true_wf map_select length-map
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin dependent_pairFormation cut hypothesisEquality applyEquality introduction extract_by_obid isectElimination natural_numberEquality cumulativity hypothesis sqequalRule lambdaEquality functionExtensionality independent_isectElimination because_Cache imageElimination imageMemberEquality baseClosed equalityTransitivity equalitySymmetry independent_functionElimination dependent_functionElimination unionElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll setElimination rename productEquality dependent_set_memberEquality applyLambdaEquality functionEquality universeEquality

Latex:
\mforall{}[A,B:Type].    \mforall{}f:A  {}\mrightarrow{}  B.  \mforall{}as,bs:A  List.    (Inj(A;B;f)  {}\mRightarrow{}  (as  \msubseteq{}  bs  \mLeftarrow{}{}\mRightarrow{}  map(f;as)  \msubseteq{}  map(f;bs)))

Date html generated: 2017_04_14-AM-09_29_53
Last ObjectModification: 2017_02_27-PM-04_02_11

Theory : list_1

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