Nuprl Lemma : subtype_rel_isect_as_subtyping_lemma
∀[A,T:Type]. ∀[B:T ⟶ Type]. A ⊆r (⋂x:T. B[x]) supposing ∀[x:T]. (A ⊆r B[x])
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
isect: ⋂x:A. B[x],
function: 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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
uall_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
isect_memberEquality,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
axiomEquality,
lemma_by_obid,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A,T:Type]. \mforall{}[B:T {}\mrightarrow{} Type]. A \msubseteq{}r (\mcap{}x:T. B[x]) supposing \mforall{}[x:T]. (A \msubseteq{}r B[x])
Date html generated:
2016_05_13-PM-03_18_53
Last ObjectModification:
2015_12_26-AM-09_08_07
Theory : subtype_0
Home
Index