Nuprl Lemma : dep-isect-subtype2
∀[A1,A2:Type]. ∀[B1:A1 ⟶ Type]. ∀[B2:A2 ⟶ Type].
(x:A1 ⋂ B1[x] ⊆r x:A2 ⋂ B2[x]) supposing ((∀x:A1. (B1[x] ⊆r B2[x])) and (A1 ⊆r A2))
Proof
Definitions occuring in Statement :
dep-isect: x:A ⋂ B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
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,
subtype_rel: A ⊆r B,
guard: {T},
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ
Lemmas referenced :
dep-isect-subtype,
subtype_rel_transitivity,
dep-isect_wf,
all_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
dependentIntersectionElimination,
sqequalHypSubstitution,
dependentIntersection_memberEquality,
hypothesisEquality,
applyEquality,
thin,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesis,
instantiate,
isectElimination,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}[A1,A2:Type]. \mforall{}[B1:A1 {}\mrightarrow{} Type]. \mforall{}[B2:A2 {}\mrightarrow{} Type].
(x:A1 \mcap{} B1[x] \msubseteq{}r x:A2 \mcap{} B2[x]) supposing ((\mforall{}x:A1. (B1[x] \msubseteq{}r B2[x])) and (A1 \msubseteq{}r A2))
Date html generated:
2018_07_25-PM-01_30_18
Last ObjectModification:
2018_06_09-PM-09_18_12
Theory : dependent!intersection
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