Nuprl Lemma : fpf-restrict_wf2
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:x:A fp-> B[x]]. ∀[P:A ⟶ 𝔹]. (fpf-restrict(f;P) ∈ x:A fp-> B[x])
Proof
Definitions occuring in Statement :
fpf-restrict: fpf-restrict(f;P),
fpf: a:A fp-> B[a],
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-restrict: fpf-restrict(f;P),
fpf: a:A fp-> B[a],
fpf-domain: fpf-domain(f),
mk_fpf: mk_fpf(L;f),
uall: ∀[x:A]. B[x],
member: t ∈ T,
pi1: fst(t),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
pi2: snd(t),
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
pi1_wf_top,
list_wf,
subtype_rel_product,
top_wf,
subtype_rel_self,
set_wf,
subtype_rel_sets,
member_filter_2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
dependent_pairEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
productElimination,
applyEquality,
lambdaEquality,
hypothesis,
setEquality,
functionEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:x:A fp-> B[x]]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. (fpf-restrict(f;P) \mmember{} x:A fp-> B[x])
Date html generated:
2018_05_21-PM-09_31_05
Last ObjectModification:
2018_02_09-AM-10_25_30
Theory : finite!partial!functions
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