Nuprl Lemma : subtype_rel_sets
∀[A,B:Type]. ∀[P:A ⟶ ℙ]. ∀[Q:B ⟶ ℙ]. ({a:A| P[a]} ⊆r {b:B| Q[b]} ) supposing ((∀a:A. (P[a] ⇒ Q[a])) and ({a:A| P[a]\000C} ⊆r B))
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],
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,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
guard: {T},
all: ∀x:A. B[x]
Lemmas referenced :
all_wf,
subtype_rel_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
applyEquality,
thin,
because_Cache,
sqequalHypSubstitution,
sqequalRule,
isect_memberFormation,
introduction,
axiomEquality,
lemma_by_obid,
isectElimination,
cumulativity,
lambdaEquality,
functionEquality,
universeEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
setElimination,
rename,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[Q:B {}\mrightarrow{} \mBbbP{}].
(\{a:A| P[a]\} \msubseteq{}r \{b:B| Q[b]\} ) supposing ((\mforall{}a:A. (P[a] {}\mRightarrow{} Q[a])) and (\{a:A| P[a]\} \msubseteq{}r B))
Date html generated:
2016_05_13-PM-03_18_47
Last ObjectModification:
2015_12_26-AM-09_08_29
Theory : subtype_0
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