Nuprl Lemma : msfun-ext-mfun
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. (mcomplete(X with d) ⇒ msfun(X;d;Y;d') ≡ FUN(X ⟶ Y))
Proof
Definitions occuring in Statement :
mcomplete: mcomplete(M),
msfun: msfun(X;d;Y;d'),
mfun: FUN(X ⟶ Y),
mk-metric-space: X with d,
metric: metric(X),
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
msfun: msfun(X;d;Y;d'),
mfun: FUN(X ⟶ Y),
is-mfun: f:FUN(X;Y),
all: ∀x:A. B[x],
so_apply: x[s],
or: P ∨ Q,
not: ¬A,
false: False,
stable: Stable{P},
uimplies: b supposing a,
guard: {T},
is-msfun: is-msfun(X;d;Y;d';f)
Lemmas referenced :
msfun_wf,
mfun_wf,
mcomplete_wf,
mk-metric-space_wf,
metric_wf,
istype-universe,
meq_wf,
is-mfun_wf,
stable__meq,
false_wf,
msep_wf,
not_wf,
istype-void,
not-msep,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
msep-not-meq,
m-strong-extensionality,
is-msfun_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
independent_pairFormation,
lambdaEquality_alt,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
applyEquality,
unionEquality,
functionEquality,
functionIsType,
independent_functionElimination,
unionIsType,
because_Cache,
independent_isectElimination,
unionElimination,
voidElimination
Latex:
\mforall{}[X,Y:Type]. \mforall{}[d:metric(X)]. \mforall{}[d':metric(Y)].
(mcomplete(X with d) {}\mRightarrow{} msfun(X;d;Y;d') \mequiv{} FUN(X {}\mrightarrow{} Y))
Date html generated:
2019_10_30-AM-06_48_17
Last ObjectModification:
2019_10_02-AM-10_59_16
Theory : reals
Home
Index