Nuprl Lemma : fpf-ap-equal
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[x:A]. ∀[v:B[x]].
(f(x) = v ∈ B[x]) supposing ((↑x ∈ dom(f)) and f || x : v)
Proof
Definitions occuring in Statement :
fpf-single: x : v,
fpf-compatible: f || g,
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],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf-compatible: f || g,
all: ∀x:A. B[x],
top: Top,
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
fpf_ap_single_lemma,
fpf-single-dom,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf-compatible_wf,
fpf-single_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
lemma_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
isectElimination,
because_Cache,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
lambdaFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[x:A]. \mforall{}[v:B[x]].
(f(x) = v) supposing ((\muparrow{}x \mmember{} dom(f)) and f || x : v)
Date html generated:
2018_05_21-PM-09_29_46
Last ObjectModification:
2018_02_09-AM-10_24_27
Theory : finite!partial!functions
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