Nuprl Lemma : sub-bag-map-equal

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


Definitions occuring in Statement :  sub-bag: sub-bag(T;as;bs) bag-map: bag-map(f;bs) bag: bag(T) uimplies: supposing a uall: [x:A]. B[x] function: x:A ⟶ B[x] universe: Type equal: t ∈ T
Definitions unfolded in proof :  sub-bag: sub-bag(T;as;bs) exists: x:A. B[x] member: t ∈ T uall: [x:A]. B[x] true: True squash: T prop: subtype_rel: A ⊆B uimplies: supposing a bag-append: as bs bag-map: bag-map(f;bs) empty-bag: {} top: Top all: x:A. B[x] guard: {T} iff: ⇐⇒ Q and: P ∧ Q implies:  Q uiff: uiff(P;Q) bag-null: bag-null(bs)
Lemmas referenced :  bag_wf bag_to_squash_list equal_wf bag-map_wf bag-append_wf list-subtype-bag map_append_sq equal-wf-T-base bag-append-empty bag-subtype-list sub-bag_wf squash_wf true_wf iff_weakening_equal append_assoc map_wf subtype_rel_list top_wf bag-append-cancel nil_wf append_wf bag-append-eq-empty assert-bag-null bag-map-null
Rules used in proof :  sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity productElimination thin cut introduction extract_by_obid isectElimination hypothesisEquality hypothesis equalityTransitivity equalitySymmetry because_Cache natural_numberEquality imageElimination promote_hyp hyp_replacement applyLambdaEquality cumulativity functionExtensionality applyEquality rename independent_isectElimination lambdaEquality sqequalRule isect_memberEquality voidElimination voidEquality baseClosed dependent_functionElimination functionEquality universeEquality isect_memberFormation axiomEquality imageMemberEquality independent_functionElimination equalityElimination independent_pairFormation

\mforall{}[T,U:Type].  \mforall{}[b1,b2:bag(T)].  \mforall{}[f:T  {}\mrightarrow{}  U].
    (b1  =  b2)  supposing  (sub-bag(T;b2;b1)  and  sub-bag(U;bag-map(f;b1);bag-map(f;b2)))

Date html generated: 2017_10_01-AM-09_05_16
Last ObjectModification: 2017_07_26-PM-04_45_20

Theory : bags

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