Nuprl Lemma : fpf-ap_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[eq:EqDecider(A)]. ∀[x:A]. f(x) ∈ B[x] supposing ↑x ∈ dom(f)
Proof
Definitions occuring in Statement :
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
pi2: snd(t),
pi1: fst(t),
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q
Lemmas referenced :
assert_wf,
deq-member_wf,
pi1_wf_top,
list_wf,
subtype_rel_product,
l_member_wf,
top_wf,
deq_wf,
assert-deq-member
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
applyEquality,
productElimination,
thin,
sqequalHypSubstitution,
hypothesisEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
isectElimination,
lambdaEquality,
functionEquality,
setEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_set_memberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[eq:EqDecider(A)]. \mforall{}[x:A].
f(x) \mmember{} B[x] supposing \muparrow{}x \mmember{} dom(f)
Date html generated:
2018_05_21-PM-09_17_49
Last ObjectModification:
2018_02_09-AM-10_16_44
Theory : finite!partial!functions
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