Nuprl Lemma : subtype_rel_isect_general
∀[A,T:Type]. ∀[C:A ⟶ Type]. ∀[B:T ⟶ Type]. (⋂x:A. C[x]) ⊆r (⋂x:T. B[x]) supposing (T ⊆r A) ∧ (∀x:T. (C[x] ⊆r B[x]))
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
and: P ∧ Q,
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,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x]
Lemmas referenced :
equal_wf,
subtype_rel_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lambdaEquality,
isect_memberEquality,
hypothesisEquality,
applyEquality,
sqequalRule,
isectElimination,
hypothesis,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
cumulativity,
lambdaFormation,
dependent_functionElimination,
extract_by_obid,
independent_functionElimination,
isectEquality,
axiomEquality,
productEquality,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}[A,T:Type]. \mforall{}[C:A {}\mrightarrow{} Type]. \mforall{}[B:T {}\mrightarrow{} Type].
(\mcap{}x:A. C[x]) \msubseteq{}r (\mcap{}x:T. B[x]) supposing (T \msubseteq{}r A) \mwedge{} (\mforall{}x:T. (C[x] \msubseteq{}r B[x]))
Date html generated:
2017_04_14-AM-07_14_02
Last ObjectModification:
2017_02_27-PM-02_49_44
Theory : subtype_0
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