Nuprl Lemma : singleton-type-function
∀[A:Type]. ∀[B:A ⟶ Type]. ((∀a:A. singleton-type(B[a])) ⇒ singleton-type(a:A ⟶ B[a]))
Proof
Definitions occuring in Statement :
singleton-type: singleton-type(A),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
singleton-type: singleton-type(A),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
guard: {T},
pi1: fst(t)
Lemmas referenced :
all_wf,
equal_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
promote_hyp,
thin,
sqequalHypSubstitution,
productElimination,
dependent_pairFormation,
hypothesisEquality,
functionExtensionality,
functionEquality,
applyEquality,
lemma_by_obid,
isectElimination,
lambdaEquality,
cumulativity,
universeEquality,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. ((\mforall{}a:A. singleton-type(B[a])) {}\mRightarrow{} singleton-type(a:A {}\mrightarrow{} B[a]))
Date html generated:
2016_05_14-PM-04_02_12
Last ObjectModification:
2015_12_26-PM-07_43_08
Theory : equipollence!!cardinality!
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