Nuprl Lemma : fpf-rename-dom
∀[A,C:Type]. ∀[B:A ⟶ Type].
∀eqa:EqDecider(A). ∀eqc:EqDecider(C). ∀r:A ⟶ C. ∀f:a:A fp-> B[a]. ∀c:C.
(↑c ∈ dom(rename(r;f)) ⇐⇒ ∃a:A. ((↑a ∈ dom(f)) c∧ (c = (r a) ∈ C)))
Proof
Definitions occuring in Statement :
fpf-rename: rename(r;f),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
fpf-dom: x ∈ dom(f),
fpf-rename: rename(r;f),
fpf: a:A fp-> B[a],
pi1: fst(t),
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
exists_wf,
l_member_wf,
equal_wf,
member_map,
map_wf,
iff_wf,
assert_wf,
deq-member_wf,
assert-deq-member,
cand_functionality_wrt_iff,
iff_weakening_equal,
fpf_wf,
deq_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
lambdaFormation,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
addLevel,
productElimination,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
isect_memberFormation,
existsFunctionality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
existsLevelFunctionality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,C:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}eqa:EqDecider(A). \mforall{}eqc:EqDecider(C). \mforall{}r:A {}\mrightarrow{} C. \mforall{}f:a:A fp-> B[a]. \mforall{}c:C.
(\muparrow{}c \mmember{} dom(rename(r;f)) \mLeftarrow{}{}\mRightarrow{} \mexists{}a:A. ((\muparrow{}a \mmember{} dom(f)) c\mwedge{} (c = (r a))))
Date html generated:
2018_05_21-PM-09_26_45
Last ObjectModification:
2018_02_09-AM-10_22_09
Theory : finite!partial!functions
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