### Nuprl Lemma : bag-map-no-repeats

`∀[T1,T2:Type]. ∀[f:T1 ⟶ T2]. ∀[bs:bag(T1)].`
`  uiff(bag-no-repeats(T2;bag-map(f;bs));bag-no-repeats(T1;bs)) supposing Inj(T1;T2;f)`

Proof

Definitions occuring in Statement :  bag-no-repeats: `bag-no-repeats(T;bs)` bag-map: `bag-map(f;bs)` bag: `bag(T)` inject: `Inj(A;B;f)` uiff: `uiff(P;Q)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  bag-no-repeats: `bag-no-repeats(T;bs)` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` uiff: `uiff(P;Q)` and: `P ∧ Q` squash: `↓T` exists: `∃x:A. B[x]` prop: `ℙ` bag-map: `bag-map(f;bs)` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` bag: `bag(T)` quotient: `x,y:A//B[x; y]` cand: `A c∧ B` all: `∀x:A. B[x]` implies: `P `` Q` iff: `P `⇐⇒` Q` no_repeats: `no_repeats(T;l)` top: `Top` not: `¬A` false: `False` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` true: `True` guard: `{T}` rev_implies: `P `` Q` inject: `Inj(A;B;f)`
Lemmas referenced :  bag_to_squash_list equal_wf bag_wf bag-map_wf squash_wf exists_wf list_wf list-subtype-bag no_repeats_wf inject_wf no_repeats_functionality_wrt_permutation length-map select_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf not_wf nat_wf less_than_wf length_wf member_wf map_wf permutation_wf subtype_rel_list top_wf lelt_wf true_wf iff_weakening_equal select-map no_repeats_map subtype_rel_dep_function l_member_wf set_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut independent_pairFormation sqequalHypSubstitution imageElimination productElimination thin extract_by_obid isectElimination hypothesisEquality promote_hyp hypothesis equalitySymmetry hyp_replacement applyLambdaEquality cumulativity equalityTransitivity functionExtensionality applyEquality rename lambdaEquality productEquality because_Cache independent_isectElimination imageMemberEquality baseClosed independent_pairEquality isect_memberEquality functionEquality universeEquality pertypeElimination dependent_pairFormation dependent_functionElimination independent_functionElimination voidElimination voidEquality lambdaFormation setElimination natural_numberEquality unionElimination int_eqEquality intEquality computeAll dependent_set_memberEquality setEquality

Latex:
\mforall{}[T1,T2:Type].  \mforall{}[f:T1  {}\mrightarrow{}  T2].  \mforall{}[bs:bag(T1)].
uiff(bag-no-repeats(T2;bag-map(f;bs));bag-no-repeats(T1;bs))  supposing  Inj(T1;T2;f)

Date html generated: 2017_10_01-AM-08_51_51
Last ObjectModification: 2017_07_26-PM-04_33_35

Theory : bags

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