Nuprl Lemma : subtype_rel_isect-2
∀[A:Type]. ∀[B1,B2:A ⟶ Type]. (⋂x:A. B1[x]) ⊆r (⋂x:A. B2[x]) supposing ∀[x:A]. (B1[x] ⊆r B2[x])
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
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,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
prop: ℙ
Lemmas referenced :
subtype_rel_isect_general,
uall_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
hypothesis,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B1,B2:A {}\mrightarrow{} Type]. (\mcap{}x:A. B1[x]) \msubseteq{}r (\mcap{}x:A. B2[x]) supposing \mforall{}[x:A]. (B1[x] \msubseteq{}r B2[x])
Date html generated:
2016_05_13-PM-03_18_54
Last ObjectModification:
2015_12_26-AM-09_08_06
Theory : subtype_0
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