### Nuprl Lemma : sv-bag-only-map

`∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[b:bag(A)].`
`  (sv-bag-only(bag-map(f;b)) = f[sv-bag-only(b)] ∈ B) supposing (0 < #(b) and single-valued-bag(b;A))`

Proof

Definitions occuring in Statement :  sv-bag-only: `sv-bag-only(b)` single-valued-bag: `single-valued-bag(b;T)` bag-size: `#(bs)` bag-map: `bag-map(f;bs)` bag: `bag(T)` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` squash: `↓T` prop: `ℙ` so_lambda: `λ2x.t[x]` uimplies: `b supposing a` subtype_rel: `A ⊆r B` top: `Top` so_apply: `x[s]` implies: `P `` Q` sq_stable: `SqStable(P)` exists: `∃x:A. B[x]` bag-size: `#(bs)` sv-bag-only: `sv-bag-only(b)` bag-map: `bag-map(f;bs)` nat: `ℕ` listp: `A List+` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` and: `P ∧ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  bag_to_squash_list sq_stable__uall single-valued-bag_wf isect_wf less_than_wf bag-size_wf equal_wf sv-bag-only_wf bag-map_wf single-valued-bag-map bag-size-map sq_stable__equal squash_wf list-subtype-bag nat_wf bag_wf length_wf hd_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf hd_map iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality imageElimination cumulativity hypothesis sqequalRule lambdaEquality natural_numberEquality applyEquality because_Cache functionExtensionality independent_isectElimination isect_memberEquality voidElimination voidEquality independent_functionElimination dependent_functionElimination axiomEquality equalityTransitivity equalitySymmetry productElimination promote_hyp rename setElimination hyp_replacement applyLambdaEquality imageMemberEquality baseClosed functionEquality universeEquality dependent_set_memberEquality unionElimination dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  B].  \mforall{}[b:bag(A)].
(sv-bag-only(bag-map(f;b))  =  f[sv-bag-only(b)])  supposing  (0  <  \#(b)  and  single-valued-bag(b;A))

Date html generated: 2017_10_01-AM-08_56_04
Last ObjectModification: 2017_07_26-PM-04_38_06

Theory : bags

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