Nuprl Lemma : fpf-single-dom
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[x,y:A]. ∀[v:Top]. uiff(↑x ∈ dom(y : v);x = y ∈ A)
Proof
Definitions occuring in Statement :
fpf-single: x : v,
fpf-dom: x ∈ dom(f),
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
top: Top,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
or: P ∨ Q,
fpf-single: x : v,
fpf-dom: x ∈ dom(f),
pi1: fst(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
rev_implies: P ⇐ Q,
eqof: eqof(d),
false: False
Lemmas referenced :
assert_wf,
fpf-dom_wf,
fpf-single_wf,
top_wf,
equal_wf,
deq_wf,
assert_witness,
bor_wf,
eqof_wf,
bfalse_wf,
or_wf,
false_wf,
uiff_wf,
deq_member_cons_lemma,
deq_member_nil_lemma,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
safe-assert-deq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
instantiate,
sqequalRule,
lambdaEquality,
hypothesis,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
universeEquality,
isect_memberFormation,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
dependent_functionElimination,
addLevel,
independent_pairFormation,
independent_isectElimination,
lambdaFormation,
orFunctionality,
unionElimination,
rename,
inlFormation
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[x,y:A]. \mforall{}[v:Top]. uiff(\muparrow{}x \mmember{} dom(y : v);x = y)
Date html generated:
2018_05_21-PM-09_29_01
Last ObjectModification:
2018_02_09-AM-10_24_06
Theory : finite!partial!functions
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