Nuprl Lemma : double_isect_subtype_rel
∀[T1,T2:Type]. ∀[B:T1 ⟶ T2 ⟶ Type]. ∀[x:T1]. ∀[y:T2]. ((⋂x:T1. ⋂y:T2. B[x;y]) ⊆r B[x;y])
Proof
Definitions occuring in Statement :
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,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
hypothesis,
thin,
isectEquality,
cumulativity,
applyEquality,
functionExtensionality,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[B:T1 {}\mrightarrow{} T2 {}\mrightarrow{} Type]. \mforall{}[x:T1]. \mforall{}[y:T2]. ((\mcap{}x:T1. \mcap{}y:T2. B[x;y]) \msubseteq{}r B[x;y])
Date html generated:
2017_04_14-AM-07_14_04
Last ObjectModification:
2017_02_27-PM-02_49_57
Theory : subtype_0
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