### 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)))`

Proof

Definitions occuring in Statement :  sub-bag: `sub-bag(T;as;bs)` bag-map: `bag-map(f;bs)` bag: `bag(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = 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: `ℙ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` bag-append: `as + bs` bag-map: `bag-map(f;bs)` empty-bag: `{}` top: `Top` all: `∀x:A. B[x]` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` 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

Latex:
\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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