Nuprl Lemma : fpf-trivial-subtype-set
∀[A:Type]. ∀[P:A ⟶ ℙ]. ∀[f:a:{a:A| P[a]} fp-> Type × Top]. (f ∈ a:A fp-> Type × Top)
Proof
Definitions occuring in Statement :
fpf: a:A fp-> B[a],
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: 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],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
fpf: a:A fp-> B[a]
Lemmas referenced :
subtype-fpf3,
top_wf,
strong-subtype-set2,
subtype_rel_self,
set_wf,
l_member_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
cumulativity,
setEquality,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
universeEquality,
independent_isectElimination,
lambdaFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[f:a:\{a:A| P[a]\} fp-> Type \mtimes{} Top]. (f \mmember{} a:A fp-> Type \mtimes{} Top)
Date html generated:
2019_10_16-AM-11_25_09
Last ObjectModification:
2018_08_22-AM-09_57_31
Theory : finite!partial!functions
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