Nuprl Lemma : isect_subtype
∀[A1:Type]. ∀[B1:A1 ⟶ ℙ]. ∀[A2:Type]. ∀[B2:A2 ⟶ ℙ].
((⋂x:A1. B1[x]) ⊆r (⋂x:A2. B2[x])) supposing ((∀x:A2. (B1[x] ⊆r B2[x])) and (A2 ⊆r A1))
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q
Lemmas referenced :
subtype_rel_isect_general,
all_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
universeEquality,
because_Cache,
independent_isectElimination,
independent_pairFormation,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity
Latex:
\mforall{}[A1:Type]. \mforall{}[B1:A1 {}\mrightarrow{} \mBbbP{}]. \mforall{}[A2:Type]. \mforall{}[B2:A2 {}\mrightarrow{} \mBbbP{}].
((\mcap{}x:A1. B1[x]) \msubseteq{}r (\mcap{}x:A2. B2[x])) supposing ((\mforall{}x:A2. (B1[x] \msubseteq{}r B2[x])) and (A2 \msubseteq{}r A1))
Date html generated:
2016_05_13-PM-04_10_39
Last ObjectModification:
2015_12_26-AM-11_21_56
Theory : subtype_1
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