Nuprl Lemma : member-fset-map
∀[T:Type]. ∀[eqT:EqDecider(T)]. ∀[X:Type]. ∀[eqX:EqDecider(X)]. ∀[f:T ⟶ X]. ∀[s:fset(T)]. ∀[x:X].
uiff(x ∈ fset-map(f;s);↓∃y:T. (y ∈ s ∧ (x = (f y) ∈ X)))
Proof
Definitions occuring in Statement :
fset-map: fset-map(f;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
fset: fset(T),
quotient: x,y:A//B[x; y],
fset-map: fset-map(f;s),
fset-member: a ∈ s,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
sq_type: SQType(T),
true: True
Lemmas referenced :
fset-member_wf,
fset-map_wf,
fset-member_witness,
squash_wf,
equal_wf,
fset_wf,
deq_wf,
istype-universe,
assert-deq-member,
map_wf,
member-map,
subtype_rel_self,
list_wf,
set-equal_wf,
decidable__fset-member,
set-equal-reflex,
l_member_wf,
subtype_base_sq,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
Error :universeIsType,
extract_by_obid,
isectElimination,
independent_functionElimination,
productEquality,
applyEquality,
productElimination,
independent_pairEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
because_Cache,
Error :functionIsType,
instantiate,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
Error :dependent_pairFormation_alt,
Error :productIsType,
Error :equalityIstype,
sqequalBase,
unionElimination,
promote_hyp,
Error :lambdaFormation_alt,
pointwiseFunctionality,
voidElimination,
Error :equalityIsType4,
intEquality,
natural_numberEquality,
cumulativity,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eqT:EqDecider(T)]. \mforall{}[X:Type]. \mforall{}[eqX:EqDecider(X)]. \mforall{}[f:T {}\mrightarrow{} X]. \mforall{}[s:fset(T)]. \mforall{}[x:X].
uiff(x \mmember{} fset-map(f;s);\mdownarrow{}\mexists{}y:T. (y \mmember{} s \mwedge{} (x = (f y))))
Date html generated:
2019_06_20-PM-01_58_53
Last ObjectModification:
2018_11_23-PM-04_42_10
Theory : finite!sets
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