Nuprl Lemma : assert-fpf-is-empty
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:x:A fp-> B[x]].  uiff(↑fpf-is-empty(f);f = ⊗ ∈ x:A fp-> B[x])
Proof
Definitions occuring in Statement : 
fpf-is-empty: fpf-is-empty(f)
, 
fpf-empty: ⊗
, 
fpf: a:A fp-> B[a]
, 
assert: ↑b
, 
uiff: uiff(P;Q)
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
fpf: a:A fp-> B[a]
, 
fpf-is-empty: fpf-is-empty(f)
, 
pi1: fst(t)
, 
fpf-empty: ⊗
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
rev_uimplies: rev_uimplies(P;Q)
, 
uimplies: b supposing a
, 
squash: ↓T
, 
prop: ℙ
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
all: ∀x:A. B[x]
, 
top: Top
, 
true: True
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
false: False
Lemmas referenced : 
eq_int_wf, 
length_wf, 
assert_of_eq_int, 
equal_wf, 
squash_wf, 
true_wf, 
length_of_null_list, 
nil_wf, 
and_wf, 
list_wf, 
l_member_wf, 
pi1_wf_top, 
subtype_rel_product, 
top_wf, 
iff_weakening_equal, 
assert_wf, 
assert_witness, 
equal-wf-T-base, 
fpf_wf, 
length_zero, 
null_nil_lemma, 
btrue_wf, 
member-implies-null-eq-bfalse, 
null_wf3, 
subtype_rel_list, 
btrue_neq_bfalse, 
set_wf
Rules used in proof : 
sqequalHypSubstitution, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
productElimination, 
thin, 
sqequalRule, 
cut, 
introduction, 
extract_by_obid, 
isectElimination, 
cumulativity, 
hypothesisEquality, 
hypothesis, 
natural_numberEquality, 
independent_isectElimination, 
applyEquality, 
lambdaEquality, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
because_Cache, 
intEquality, 
dependent_set_memberEquality, 
independent_pairFormation, 
productEquality, 
functionEquality, 
setEquality, 
setElimination, 
rename, 
applyLambdaEquality, 
functionExtensionality, 
lambdaFormation, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
imageMemberEquality, 
baseClosed, 
universeEquality, 
independent_functionElimination, 
isect_memberFormation, 
independent_pairEquality, 
axiomEquality, 
dependent_pairEquality
Latex:
\mforall{}[A:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[f:x:A  fp->  B[x]].    uiff(\muparrow{}fpf-is-empty(f);f  =  \motimes{})
Date html generated:
2018_05_21-PM-09_17_43
Last ObjectModification:
2018_02_09-AM-10_16_42
Theory : finite!partial!functions
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