Nuprl Lemma : mfun-subtype2
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[A:Type]. FUN(X ⟶ Y) ⊆r FUN(A ⟶ Y) supposing A ⊆r X
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 :
all: ∀x:A. B[x],
is-mfun: f:FUN(X;Y),
prop: ℙ,
mfun: FUN(X ⟶ Y),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
subtype_rel_wf,
mfun_wf,
metric-on-subtype,
is-mfun_wf
Rules used in proof :
dependent_functionElimination,
lambdaFormation_alt,
because_Cache,
inhabitedIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
axiomEquality,
independent_isectElimination,
isectElimination,
extract_by_obid,
universeIsType,
sqequalRule,
hypothesis,
hypothesisEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality_alt,
rename,
thin,
setElimination,
sqequalHypSubstitution,
lambdaEquality_alt,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X,Y:Type]. \mforall{}[d:metric(X)]. \mforall{}[d':metric(Y)]. \mforall{}[A:Type].
FUN(X {}\mrightarrow{} Y) \msubseteq{}r FUN(A {}\mrightarrow{} Y) supposing A \msubseteq{}r X
Date html generated:
2019_10_30-AM-06_21_35
Last ObjectModification:
2019_10_25-PM-04_40_32
Theory : reals
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