Nuprl Lemma : fset-constrained-image-singleton
∀[eqt,eqa:Top]. ∀[T,A:Type]. ∀[f:T ⟶ A]. ∀[P:A ⟶ 𝔹]. ∀[x:T]. (f"({x}) s.t. P ~ if P (f x) then {f x} else {} fi )
Proof
Definitions occuring in Statement :
fset-constrained-image: f"(s) s.t. P,
empty-fset: {},
fset-singleton: {x},
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fset-singleton: {x},
fset-constrained-image: f"(s) s.t. P,
f-union: f-union(domeq;rngeq;s;x.g[x]),
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
fset-union: x ⋃ y,
l-union: as ⋃ bs,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
empty-fset: {}
Lemmas referenced :
list_accum_cons_lemma,
list_accum_nil_lemma,
bool_wf,
eqtt_to_assert,
reduce_cons_lemma,
reduce_nil_lemma,
insert_nil_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
lambdaFormation,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
because_Cache,
independent_functionElimination,
sqequalAxiom,
functionEquality,
universeEquality
Latex:
\mforall{}[eqt,eqa:Top]. \mforall{}[T,A:Type]. \mforall{}[f:T {}\mrightarrow{} A]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:T].
(f"(\{x\}) s.t. P \msim{} if P (f x) then \{f x\} else \{\} fi )
Date html generated:
2017_04_17-AM-09_21_11
Last ObjectModification:
2017_02_27-PM-05_23_57
Theory : finite!sets
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