### Nuprl Lemma : b_all-map

`∀[A,B:Type].`
`  ∀f:A ⟶ B. ∀b:bag(A). ∀P:B ⟶ ℙ.  ((∀b:B. SqStable(P[b])) `` (b_all(B;bag-map(f;b);x.P[x]) `⇐⇒` b_all(A;b;x.P[f x])))`

Proof

Definitions occuring in Statement :  b_all: `b_all(T;b;x.P[x])` bag-map: `bag-map(f;bs)` bag: `bag(T)` sq_stable: `SqStable(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` b_all: `b_all(T;b;x.P[x])` iff: `P `⇐⇒` Q` and: `P ∧ Q` member: `t ∈ T` exists: `∃x:A. B[x]` prop: `ℙ` squash: `↓T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtype_rel: `A ⊆r B` sq_stable: `SqStable(P)` guard: `{T}`
Lemmas referenced :  bag-member_wf equal_wf all_wf squash_wf exists_wf bag-member-map bag-map_wf iff_wf sq_stable_wf bag_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut independent_pairFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin applyEquality functionExtensionality hypothesisEquality cumulativity independent_functionElimination dependent_pairFormation productEquality introduction extract_by_obid isectElimination sqequalRule imageMemberEquality baseClosed lambdaEquality functionEquality addLevel productElimination impliesFunctionality allFunctionality independent_isectElimination allLevelFunctionality impliesLevelFunctionality because_Cache universeEquality imageElimination equalityTransitivity equalitySymmetry

Latex:
\mforall{}[A,B:Type].
\mforall{}f:A  {}\mrightarrow{}  B.  \mforall{}b:bag(A).  \mforall{}P:B  {}\mrightarrow{}  \mBbbP{}.
((\mforall{}b:B.  SqStable(P[b]))  {}\mRightarrow{}  (b\_all(B;bag-map(f;b);x.P[x])  \mLeftarrow{}{}\mRightarrow{}  b\_all(A;b;x.P[f  x])))

Date html generated: 2017_10_01-AM-08_55_13
Last ObjectModification: 2017_07_26-PM-04_37_10

Theory : bags

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