Nuprl Lemma : fpf-cap_functionality
∀[A:Type]. ∀[d1,d2:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[x:A]. ∀[z:B[x]]. (f(x)?z = 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],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
fpf-ap: f(x),
pi2: snd(t),
fpf: a:A fp-> B[a],
fpf-dom: x ∈ dom(f),
pi1: fst(t),
not: ¬A,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
false: False,
fpf-cap: f(x)?z,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
fpf-dom_wf,
subtype-fpf2,
top_wf,
bool_wf,
fpf-ap_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
fpf_ap_pair_lemma,
assert-deq-member,
l_member_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
functionExtensionality,
hypothesis,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
productElimination,
dependent_functionElimination,
independent_functionElimination,
promote_hyp,
isect_memberFormation,
unionElimination,
equalityElimination,
axiomEquality
Latex:
\mforall{}[A:Type]. \mforall{}[d1,d2:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[x:A]. \mforall{}[z:B[x]].
(f(x)?z = f(x)?z)
Date html generated:
2018_05_21-PM-09_19_31
Last ObjectModification:
2018_02_09-AM-10_17_38
Theory : finite!partial!functions
Home
Index