Nuprl Lemma : subtype_rel_nested_set2
∀[A,B:Type]. ∀[P:B ⟶ ℙ]. ∀[Q:{b:B| P[b]} ⟶ ℙ]. {b:{b:B| P[b]} | Q[b]} ⊆r A supposing {b:B| P[b] ∧ Q[b]} ⊆r A
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],
and: P ∧ Q,
set: {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_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ
Lemmas referenced :
subtype_rel_transitivity,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
sqequalRule,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
axiomEquality,
cumulativity,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[P:B {}\mrightarrow{} \mBbbP{}]. \mforall{}[Q:\{b:B| P[b]\} {}\mrightarrow{} \mBbbP{}].
\{b:\{b:B| P[b]\} | Q[b]\} \msubseteq{}r A supposing \{b:B| P[b] \mwedge{} Q[b]\} \msubseteq{}r A
Date html generated:
2016_05_13-PM-03_18_50
Last ObjectModification:
2015_12_26-AM-09_08_19
Theory : subtype_0
Home
Index