Nuprl Lemma : fpf-cap_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[eq:EqDecider(A)]. ∀[x:A]. ∀[z:B[x]]. (f(x)?z ∈ B[x])
Proof
Definitions occuring in Statement :
fpf-cap: f(x)?z,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fpf-cap: f(x)?z,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ
Lemmas referenced :
deq_wf,
fpf_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
bool_wf,
fpf-ap_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
isect_memberEquality,
isectElimination,
thin,
because_Cache,
extract_by_obid,
lambdaEquality,
functionEquality,
universeEquality,
independent_isectElimination,
lambdaFormation,
voidElimination,
voidEquality,
baseClosed,
unionElimination,
equalityElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[eq:EqDecider(A)]. \mforall{}[x:A]. \mforall{}[z:B[x]].
(f(x)?z \mmember{} B[x])
Date html generated:
2018_05_21-PM-09_17_58
Last ObjectModification:
2018_02_09-AM-10_16_49
Theory : finite!partial!functions
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