Nuprl Lemma : truncation-property
∀[X,Q:Type]. ∀f:X ⟶ ⇃(Q). ((∀x:X. ((|f| |x|) = (f x) ∈ ⇃(Q))) ∧ (∀g:⇃(X) ⟶ ⇃(Q). (g = |f| ∈ (⇃(X) ⟶ ⇃(Q)))))
Proof
Definitions occuring in Statement :
truncate-map: |f|,
truncate: |x|,
quotient: x,y:A//B[x; y],
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
truncate: |x|,
truncate-map: |f|,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a
Lemmas referenced :
quotient_wf,
true_wf,
equiv_rel_true,
istype-universe,
half-squash-equality,
truncate-map_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
sqequalRule,
applyEquality,
hypothesisEquality,
hypothesis,
Error :universeIsType,
independent_pairFormation,
Error :functionIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
Error :functionIsTypeImplies,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
instantiate,
universeEquality,
functionExtensionality
Latex:
\mforall{}[X,Q:Type]. \mforall{}f:X {}\mrightarrow{} \00D9(Q). ((\mforall{}x:X. ((|f| |x|) = (f x))) \mwedge{} (\mforall{}g:\00D9(X) {}\mrightarrow{} \00D9(Q). (g = |f|)))
Date html generated:
2019_06_20-PM-00_32_52
Last ObjectModification:
2018_11_16-AM-11_48_53
Theory : quot_1
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