Nuprl Lemma : sub-isect-family
∀[P:Type]. ∀[G:P ⟶ Type]. ∀[A:Type]. ∀[F:A ⟶ P ⟶ Type]. G ⊆ ⋂a:A. F[a] supposing ∀a:A. G ⊆ F[a]
Proof
Definitions occuring in Statement :
sub-family: F ⊆ G,
isect-family: ⋂a:A. F[a],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀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,
isect-family: ⋂a:A. F[a],
sub-family: F ⊆ G,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
all_wf,
sub-family_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaFormation,
lambdaEquality,
isect_memberEquality,
hypothesisEquality,
applyEquality,
sqequalHypSubstitution,
hypothesis,
dependent_functionElimination,
thin,
axiomEquality,
lemma_by_obid,
isectElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[P:Type]. \mforall{}[G:P {}\mrightarrow{} Type]. \mforall{}[A:Type]. \mforall{}[F:A {}\mrightarrow{} P {}\mrightarrow{} Type]. G \msubseteq{} \mcap{}a:A. F[a] supposing \mforall{}a:A. G \msubseteq{} F[a]
Date html generated:
2016_05_14-AM-06_12_10
Last ObjectModification:
2015_12_26-PM-00_06_14
Theory : co-recursion
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