### Nuprl Lemma : bag-combine-size-bound2

`∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[L:bag(A)]. ∀[a:A].  #(f[a]) ≤ #(⋃a∈L.f[a]) supposing a ↓∈ L`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-combine: `⋃x∈bs.f[x]` bag-size: `#(bs)` bag: `bag(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` squash: `↓T` so_apply: `x[s]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` sq_stable: `SqStable(P)` implies: `P `` Q` exists: `∃x:A. B[x]` prop: `ℙ` all: `∀x:A. B[x]` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` not: `¬A` false: `False`
Lemmas referenced :  bag_to_squash_list sq_stable__le bag-size_wf bag-combine_wf bag-member_wf bag-member-sq-list-member list-subtype-bag nat_wf bag-combine-size-bound le_wf less_than'_wf bag_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality imageElimination cumulativity applyEquality functionExtensionality hypothesis sqequalRule lambdaEquality independent_functionElimination productElimination promote_hyp equalitySymmetry hyp_replacement Error :applyLambdaEquality,  rename dependent_functionElimination independent_isectElimination setElimination imageMemberEquality baseClosed independent_pairEquality axiomEquality equalityTransitivity isect_memberEquality functionEquality universeEquality voidElimination

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  bag(B)].  \mforall{}[L:bag(A)].  \mforall{}[a:A].    \#(f[a])  \mleq{}  \#(\mcup{}a\mmember{}L.f[a])  supposing  a  \mdownarrow{}\mmember{}  L

Date html generated: 2016_10_25-AM-10_28_31
Last ObjectModification: 2016_07_12-AM-06_44_21

Theory : bags

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