Nuprl Lemma : fpf-compatible-single-iff
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[x:A]. ∀[v:B[x]].
  uiff(f || x : v;v = f(x) ∈ B[x] supposing ↑x ∈ dom(f))
Proof
Definitions occuring in Statement : 
fpf-single: x : v
, 
fpf-compatible: f || g
, 
fpf-ap: f(x)
, 
fpf-dom: x ∈ dom(f)
, 
fpf: a:A fp-> B[a]
, 
deq: EqDecider(T)
, 
assert: ↑b
, 
uiff: uiff(P;Q)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
prop: ℙ
, 
fpf-compatible: f || g
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
subtype_rel: A ⊆r B
, 
rev_uimplies: rev_uimplies(P;Q)
, 
top: Top
, 
cand: A c∧ B
, 
eqof: eqof(d)
, 
squash: ↓T
, 
guard: {T}
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
fpf-compatible_wf, 
fpf-single_wf, 
fpf_ap_single_lemma, 
istype-assert, 
fpf-dom_wf, 
subtype-fpf2, 
top_wf, 
fpf-ap_wf, 
fpf_wf, 
deq_wf, 
istype-universe, 
fpf-single-dom, 
fpf-single-dom-sq, 
safe-assert-deq, 
equal_wf, 
squash_wf, 
true_wf, 
assert_wf, 
subtype_rel-equal, 
subtype_rel_self, 
iff_weakening_equal, 
assert_functionality_wrt_uiff
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
independent_pairFormation, 
because_Cache, 
sqequalRule, 
sqequalHypSubstitution, 
isect_memberEquality_alt, 
isectElimination, 
thin, 
hypothesisEquality, 
axiomEquality, 
hypothesis, 
isectIsTypeImplies, 
inhabitedIsType, 
universeIsType, 
extract_by_obid, 
lambdaEquality_alt, 
applyEquality, 
instantiate, 
lambdaFormation_alt, 
dependent_functionElimination, 
Error :memTop, 
productElimination, 
productIsType, 
independent_isectElimination, 
functionIsTypeImplies, 
isectIsType, 
equalityIstype, 
independent_pairEquality, 
functionIsType, 
universeEquality, 
equalitySymmetry, 
voidEquality, 
voidElimination, 
isect_memberEquality, 
independent_functionElimination, 
imageElimination, 
equalityTransitivity, 
imageMemberEquality, 
baseClosed, 
dependent_set_memberEquality_alt, 
applyLambdaEquality, 
setElimination, 
rename, 
natural_numberEquality
Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[f:a:A  fp->  B[a]].  \mforall{}[x:A].  \mforall{}[v:B[x]].
    uiff(f  ||  x  :  v;v  =  f(x)  supposing  \muparrow{}x  \mmember{}  dom(f))
Date html generated:
2020_05_20-AM-09_03_02
Last ObjectModification:
2020_01_04-PM-11_11_37
Theory : finite!partial!functions
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