Nuprl Lemma : mfun-subtype
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[A:Type]. FUN(X ⟶ A) ⊆r FUN(X ⟶ Y) supposing A ⊆r Y
Proof
Definitions occuring in Statement :
mfun: FUN(X ⟶ Y),
metric: metric(X),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
mfun: FUN(X ⟶ Y),
is-mfun: f:FUN(X;Y),
all: ∀x:A. B[x],
prop: ℙ
Lemmas referenced :
is-mfun_wf,
mfun_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
functionExtensionality,
applyEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
universeIsType,
extract_by_obid,
isectElimination,
because_Cache,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[X,Y:Type]. \mforall{}[d:metric(X)]. \mforall{}[d':metric(Y)]. \mforall{}[A:Type].
FUN(X {}\mrightarrow{} A) \msubseteq{}r FUN(X {}\mrightarrow{} Y) supposing A \msubseteq{}r Y
Date html generated:
2019_10_30-AM-06_21_16
Last ObjectModification:
2019_10_02-AM-09_57_24
Theory : reals
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