Nuprl Lemma : set-elim
∀[A:Type]. ∀[B:A ⟶ Type]. ∀x:Image((a:A × B[a]),(λp.(fst(p)))). ((x ∈ A) ∧ (↓B[x]))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
image-type: Image(T,f),
all: ∀x:A. B[x],
squash: ↓T,
and: P ∧ Q,
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
squash: ↓T,
so_apply: x[s],
cand: A c∧ B,
and: P ∧ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
pi1: fst(t)
Lemmas referenced :
image-type_wf
Rules used in proof :
universeEquality,
Error :isect_memberEquality_alt,
Error :functionIsType,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_pairEquality,
productElimination,
dependent_functionElimination,
Error :lambdaEquality_alt,
baseClosed,
applyEquality,
hypothesisEquality,
productEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
Error :universeIsType,
hypothesis,
independent_pairFormation,
sqequalRule,
imageElimination,
Error :lambdaFormation_alt,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
rename
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}x:Image((a:A \mtimes{} B[a]),(\mlambda{}p.(fst(p)))). ((x \mmember{} A) \mwedge{} (\mdownarrow{}B[x]))
Date html generated:
2019_06_20-AM-11_13_36
Last ObjectModification:
2018_10_16-PM-01_02_28
Theory : core_2
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