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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