Nuprl Lemma : sq_exists_subtype_rel
∀[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 ⊆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],
sq_exists: ∃x:{A| B[x]},
implies: P ⇒ Q,
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,
sq_exists: ∃x:{A| B[x]},
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
prop: ℙ,
implies: P ⇒ Q
Lemmas referenced :
subtype_rel_sets,
subtype_rel_transitivity,
sq_exists_wf,
all_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
hypothesisEquality,
applyEquality,
lemma_by_obid,
isectElimination,
thin,
because_Cache,
sqequalRule,
independent_isectElimination,
hypothesis,
axiomEquality,
functionEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[Q:B {}\mrightarrow{} \mBbbP{}].
((\mexists{}a:\{A| P[a]\}) \msubseteq{}r (\mexists{}b:\{B| Q[b]\})) supposing ((\mforall{}a:A. (P[a] {}\mRightarrow{} Q[a])) and (A \msubseteq{}r B))
Date html generated:
2016_05_15-PM-06_37_53
Last ObjectModification:
2015_12_27-AM-11_54_12
Theory : general
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