Nuprl Lemma : l_disjoint-fpf-dom
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[f:a:A fp-> Top]. ∀[L:A List].
  uiff(l_disjoint(A;fst(f);L);∀[a:A]. ¬(a ∈ L) supposing ↑a ∈ dom(f))
Proof
Definitions occuring in Statement : 
fpf-dom: x ∈ dom(f)
, 
fpf: a:A fp-> B[a]
, 
l_disjoint: l_disjoint(T;l1;l2)
, 
l_member: (x ∈ l)
, 
list: T List
, 
deq: EqDecider(T)
, 
assert: ↑b
, 
uiff: uiff(P;Q)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
top: Top
, 
pi1: fst(t)
, 
not: ¬A
, 
universe: Type
Definitions unfolded in proof : 
fpf-dom: x ∈ dom(f)
, 
l_disjoint: l_disjoint(T;l1;l2)
, 
fpf: a:A fp-> B[a]
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
false: False
, 
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
, 
cand: A c∧ B
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
l_member_wf, 
assert_wf, 
deq-member_wf, 
pi1_wf_top, 
list_wf, 
subtype_rel_product, 
top_wf, 
all_wf, 
not_wf, 
and_wf, 
uall_wf, 
isect_wf, 
deq_wf, 
assert-deq-member
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation, 
introduction, 
cut, 
independent_pairFormation, 
lambdaFormation, 
thin, 
hypothesis, 
sqequalHypSubstitution, 
independent_functionElimination, 
voidElimination, 
lemma_by_obid, 
isectElimination, 
hypothesisEquality, 
lambdaEquality, 
dependent_functionElimination, 
because_Cache, 
applyEquality, 
functionEquality, 
setEquality, 
independent_isectElimination, 
isect_memberEquality, 
voidEquality, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
independent_pairEquality, 
productEquality, 
universeEquality
Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[f:a:A  fp->  Top].  \mforall{}[L:A  List].
    uiff(l\_disjoint(A;fst(f);L);\mforall{}[a:A].  \mneg{}(a  \mmember{}  L)  supposing  \muparrow{}a  \mmember{}  dom(f))
Date html generated:
2018_05_21-PM-09_31_39
Last ObjectModification:
2018_02_09-AM-10_26_36
Theory : finite!partial!functions
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