Nuprl Lemma : fpf-restrict-dom
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f:x:A fp-> B[x]]. ∀[P:A ⟶ 𝔹]. ∀[x:A].
  uiff(↑x ∈ dom(fpf-restrict(f;P));{(↑x ∈ dom(f)) ∧ (↑(P x))})
Proof
Definitions occuring in Statement : 
fpf-restrict: fpf-restrict(f;P)
, 
fpf-dom: x ∈ dom(f)
, 
fpf: a:A fp-> B[a]
, 
deq: EqDecider(T)
, 
assert: ↑b
, 
bool: 𝔹
, 
uiff: uiff(P;Q)
, 
uall: ∀[x:A]. B[x]
, 
guard: {T}
, 
so_apply: x[s]
, 
and: P ∧ Q
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
iff: P 
⇐⇒ Q
, 
uimplies: b supposing a
, 
implies: P 
⇒ Q
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
prop: ℙ
, 
rev_implies: P 
⇐ Q
, 
top: Top
Lemmas referenced : 
assert_witness, 
fpf-dom_wf, 
assert_wf, 
fpf-restrict_wf2, 
top_wf, 
bool_wf, 
fpf_wf, 
deq_wf, 
domain_fpf_restrict_lemma, 
and_wf, 
l_member_wf, 
fpf-domain_wf, 
subtype-fpf2, 
member_filter, 
filter_wf5, 
subtype_rel_dep_function, 
subtype_rel_self, 
set_wf, 
iff_wf, 
member-fpf-domain
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
thin, 
sqequalHypSubstitution, 
dependent_functionElimination, 
hypothesisEquality, 
hypothesis, 
independent_pairFormation, 
productElimination, 
independent_functionElimination, 
sqequalRule, 
independent_pairEquality, 
lemma_by_obid, 
isectElimination, 
applyEquality, 
because_Cache, 
lambdaEquality, 
isect_memberEquality, 
equalityTransitivity, 
equalitySymmetry, 
functionEquality, 
cumulativity, 
universeEquality, 
lambdaFormation, 
voidElimination, 
voidEquality, 
independent_isectElimination, 
addLevel, 
impliesFunctionality, 
setEquality, 
setElimination, 
rename, 
andLevelFunctionality
Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[f:x:A  fp->  B[x]].  \mforall{}[P:A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[x:A].
    uiff(\muparrow{}x  \mmember{}  dom(fpf-restrict(f;P));\{(\muparrow{}x  \mmember{}  dom(f))  \mwedge{}  (\muparrow{}(P  x))\})
Date html generated:
2018_05_21-PM-09_31_20
Last ObjectModification:
2018_02_09-AM-10_25_45
Theory : finite!partial!functions
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