Nuprl Lemma : subtype_rel_double_isect
∀[A,T1,T2:Type]. ∀[B:T1 ⟶ T2 ⟶ Type]. uiff(A ⊆r (⋂x:T1. ⋂y:T2. B[x;y]);∀[x:T1]. ∀[y:T2]. (A ⊆r B[x;y]))
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
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[s1;s2],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Lemmas referenced :
subtype_rel_wf,
uall_wf,
subtype_rel_transitivity,
equal_wf
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,
extract_by_obid,
cumulativity,
isectEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[A,T1,T2:Type]. \mforall{}[B:T1 {}\mrightarrow{} T2 {}\mrightarrow{} Type].
uiff(A \msubseteq{}r (\mcap{}x:T1. \mcap{}y:T2. B[x;y]);\mforall{}[x:T1]. \mforall{}[y:T2]. (A \msubseteq{}r B[x;y]))
Date html generated:
2017_04_14-AM-07_14_03
Last ObjectModification:
2017_02_27-PM-02_49_46
Theory : subtype_0
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