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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