Nuprl Lemma : sub-bag-combine

[T,U:Type].  ∀f:T ⟶ bag(U). ∀b1,b2:bag(T).  (sub-bag(T;b1;b2)  sub-bag(U;⋃x∈b1.f[x];⋃x∈b2.f[x]))


Proof




Definitions occuring in Statement :  sub-bag: sub-bag(T;as;bs) bag-combine: x∈bs.f[x] bag: bag(T) uall: [x:A]. B[x] so_apply: x[s] all: x:A. B[x] implies:  Q function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] all: x:A. B[x] implies:  Q sub-bag: sub-bag(T;as;bs) exists: x:A. B[x] member: t ∈ T squash: T prop: so_lambda: λ2x.t[x] so_apply: x[s] true: True subtype_rel: A ⊆B uimplies: supposing a guard: {T} iff: ⇐⇒ Q and: P ∧ Q rev_implies:  Q
Lemmas referenced :  bag-combine-append-left sub-bag_wf squash_wf true_wf bag_wf bag-combine_wf iff_weakening_equal equal_wf bag-append_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin cut hypothesis introduction extract_by_obid isectElimination hypothesisEquality applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry because_Cache cumulativity sqequalRule functionExtensionality natural_numberEquality imageMemberEquality baseClosed universeEquality independent_isectElimination independent_functionElimination dependent_pairFormation hyp_replacement Error :applyLambdaEquality,  functionEquality

Latex:
\mforall{}[T,U:Type].    \mforall{}f:T  {}\mrightarrow{}  bag(U).  \mforall{}b1,b2:bag(T).    (sub-bag(T;b1;b2)  {}\mRightarrow{}  sub-bag(U;\mcup{}x\mmember{}b1.f[x];\mcup{}x\mmember{}b2.f[x]))



Date html generated: 2016_10_25-AM-10_26_28
Last ObjectModification: 2016_07_12-AM-06_42_20

Theory : bags


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