Nuprl Lemma : member-fpf-domain-variant
∀[A,V:Type]. ∀f:a:A fp-> V × Top. ∀eq:EqDecider(A). ∀x:A. (↑x ∈ dom(f) ⇐⇒ (x ∈ fpf-domain(f)))
Proof
Definitions occuring in Statement :
fpf-domain: fpf-domain(f),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
l_member: (x ∈ l),
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
fpf: a:A fp-> B[a],
fpf-domain: fpf-domain(f),
fpf-dom: x ∈ dom(f),
pi1: fst(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
l_member_wf,
assert-deq-member,
assert_wf,
deq-member_wf,
iff_wf,
deq_wf,
fpf_wf,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
independent_pairFormation,
hypothesis,
lemma_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
addLevel,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality,
productEquality,
universeEquality
Latex:
\mforall{}[A,V:Type]. \mforall{}f:a:A fp-> V \mtimes{} Top. \mforall{}eq:EqDecider(A). \mforall{}x:A. (\muparrow{}x \mmember{} dom(f) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} fpf-domain(f)))
Date html generated:
2018_05_21-PM-09_17_21
Last ObjectModification:
2018_02_09-AM-10_16_30
Theory : finite!partial!functions
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