Nuprl Lemma : fpf-all-empty
∀[A:Type]. ∀eq,P:Top.  (∀y∈dom(⊗). w=⊗(y) 
⇒  P[y;w] 
⇐⇒ True)
Proof
Definitions occuring in Statement : 
fpf-all: ∀x∈dom(f). v=f(x) 
⇒  P[x; v]
, 
fpf-empty: ⊗
, 
uall: ∀[x:A]. B[x]
, 
top: Top
, 
so_apply: x[s1;s2]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
true: True
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
fpf-empty: ⊗
, 
fpf-all: ∀x∈dom(f). v=f(x) 
⇒  P[x; v]
, 
member: t ∈ T
, 
fpf-dom: x ∈ dom(f)
, 
pi1: fst(t)
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
true: True
, 
false: False
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
fpf_ap_pair_lemma, 
deq_member_nil_lemma, 
istype-top, 
istype-universe, 
istype-void, 
istype-true
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
sqequalRule, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
Error :memTop, 
hypothesis, 
inhabitedIsType, 
hypothesisEquality, 
instantiate, 
isectElimination, 
universeEquality, 
independent_pairFormation, 
natural_numberEquality, 
functionIsType, 
universeIsType, 
voidElimination, 
because_Cache
Latex:
\mforall{}[A:Type].  \mforall{}eq,P:Top.    (\mforall{}y\mmember{}dom(\motimes{}).  w=\motimes{}(y)  {}\mRightarrow{}    P[y;w]  \mLeftarrow{}{}\mRightarrow{}  True)
Date html generated:
2020_05_20-AM-09_03_22
Last ObjectModification:
2020_01_28-PM-03_38_58
Theory : finite!partial!functions
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