Nuprl Lemma : subtype_rel_list_set
∀[A,B:Type]. ∀[P:A ⟶ Type]. ∀[Q:B ⟶ Type].
(({a:A| P[a]} List) ⊆r ({b:B| Q[b]} List)) supposing ((∀x:A. (P[x] ⇒ Q[x])) and ({a:A| P[a]} ⊆r B))
Proof
Definitions occuring in Statement :
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q
Lemmas referenced :
subtype_rel_list,
subtype_rel_sets,
all_wf,
subtype_rel_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalHypSubstitution,
sqequalRule,
because_Cache,
isect_memberFormation,
introduction,
lemma_by_obid,
isectElimination,
thin,
setEquality,
independent_isectElimination,
lambdaEquality,
universeEquality,
axiomEquality,
cumulativity,
functionEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A,B:Type]. \mforall{}[P:A {}\mrightarrow{} Type]. \mforall{}[Q:B {}\mrightarrow{} Type].
((\{a:A| P[a]\} List) \msubseteq{}r (\{b:B| Q[b]\} List)) supposing ((\mforall{}x:A. (P[x] {}\mRightarrow{} Q[x])) and (\{a:A| P[a]\} \msubseteq{}\000Cr B))
Date html generated:
2016_05_14-AM-06_25_52
Last ObjectModification:
2015_12_26-PM-00_42_46
Theory : list_0
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