Nuprl Lemma : fpf-dom_functionality
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[eq1,eq2:EqDecider(A)]. ∀[f:a:A fp-> B[a]]. ∀[x:A]. x ∈ dom(f) = x ∈ dom(f)
Proof
Definitions occuring in Statement :
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
true: True,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
deq-member_wf,
bool_wf,
eqtt_to_assert,
assert-deq-member,
iff_imp_equal_bool,
btrue_wf,
true_wf,
l_member_wf,
assert_wf,
iff_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
false_wf,
fpf_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
natural_numberEquality,
addLevel,
impliesFunctionality,
because_Cache,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
isect_memberEquality,
axiomEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eq1,eq2:EqDecider(A)]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[x:A].
x \mmember{} dom(f) = x \mmember{} dom(f)
Date html generated:
2018_05_21-PM-09_17_30
Last ObjectModification:
2018_02_09-AM-10_16_32
Theory : finite!partial!functions
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