Nuprl Lemma : subtype_rel_isect
∀[A,T:Type]. ∀[B:T ⟶ Type]. uiff(A ⊆r (⋂x:T. B[x]);∀[x:T]. (A ⊆r B[x]))
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
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,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x]
Lemmas referenced :
subtype_rel_wf,
uall_wf,
subtype_rel_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
axiomEquality,
hypothesis,
hypothesisEquality,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
because_Cache,
lemma_by_obid,
isectEquality,
applyEquality,
lambdaEquality,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
independent_isectElimination
Latex:
\mforall{}[A,T:Type]. \mforall{}[B:T {}\mrightarrow{} Type]. uiff(A \msubseteq{}r (\mcap{}x:T. B[x]);\mforall{}[x:T]. (A \msubseteq{}r B[x]))
Date html generated:
2016_05_13-PM-03_18_52
Last ObjectModification:
2015_12_26-AM-09_08_08
Theory : subtype_0
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