Nuprl Lemma : id-fun-set
∀[A:Type]. ∀[P:A ⟶ ℙ]. ∀[f:id-fun(A)]. (f ∈ id-fun({a:A| P[a]} ))
Proof
Definitions occuring in Statement :
id-fun: id-fun(T),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x]
Lemmas referenced :
id-fun-subtype,
strong-subtype-set2,
id-fun_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
applyEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesis,
lambdaEquality,
sqequalRule,
universeEquality,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
functionEquality,
cumulativity
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[f:id-fun(A)]. (f \mmember{} id-fun(\{a:A| P[a]\} ))
Date html generated:
2016_05_13-PM-04_12_23
Last ObjectModification:
2015_12_26-AM-11_12_09
Theory : subtype_1
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