### Nuprl Lemma : bag-settype

`∀[T:Type]. ∀[bs:bag(T)]. ∀[P:T ⟶ ℙ].  bs ∈ bag({x:T| P[x]} ) supposing ∀x:T. (x ↓∈ bs `` P[x])`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag: `bag(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` bag: `bag(T)` all: `∀x:A. B[x]` prop: `ℙ` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` less_than: `a < b` squash: `↓T` subtype_rel: `A ⊆r B` true: `True` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]`
Lemmas referenced :  list_wf permutation_wf permutation_weakening list-set-type2 select_wf int_seg_properties length_wf decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_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_wf decidable__lt intformless_wf int_formula_prop_less_lemma bag-member-select int_seg_wf equal-wf-base member_wf squash_wf true_wf bag_wf all_wf bag-member_wf list-subtype-bag subtype_rel_self permutation-strong-subtype strong-subtype-set2 quotient-member-eq permutation-equiv
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution extract_by_obid isectElimination thin hypothesisEquality hypothesis promote_hyp lambdaFormation equalityTransitivity equalitySymmetry because_Cache dependent_functionElimination independent_isectElimination pointwiseFunctionality sqequalRule pertypeElimination productElimination lambdaEquality applyEquality cumulativity setElimination rename natural_numberEquality unionElimination approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation imageElimination productEquality setEquality imageMemberEquality baseClosed axiomEquality functionEquality universeEquality instantiate

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].    bs  \mmember{}  bag(\{x:T|  P[x]\}  )  supposing  \mforall{}x:T.  (x  \mdownarrow{}\mmember{}  bs  {}\mRightarrow{}  P[x])

Date html generated: 2018_05_21-PM-06_25_07
Last ObjectModification: 2018_05_19-PM-05_15_36

Theory : bags

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