Nuprl Lemma : discrete-presheaf-term-at-morph
∀[C:SmallCategory]. ∀[T:Type]. ∀[X:ps_context{j:l}(C)]. ∀[t:{X ⊢ _:discr(T)}].
  ∀I,J:cat-ob(C). ∀f:cat-arrow(C) J I. ∀a:X(I).  (t(a) = t(f(a)) ∈ T)
Proof
Definitions occuring in Statement : 
discrete-presheaf-type: discr(T)
, 
presheaf-term-at: u(a)
, 
presheaf-term: {X ⊢ _:A}
, 
psc-restriction: f(s)
, 
I_set: A(I)
, 
ps_context: __⊢
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
apply: f a
, 
universe: Type
, 
equal: s = t ∈ T
, 
cat-arrow: cat-arrow(C)
, 
cat-ob: cat-ob(C)
, 
small-category: SmallCategory
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
presheaf-term: {X ⊢ _:A}
, 
discrete-presheaf-type: discr(T)
, 
presheaf-term-at: u(a)
, 
guard: {T}
Lemmas referenced : 
presheaf_type_at_pair_lemma, 
presheaf_type_ap_morph_pair_lemma, 
I_set_wf, 
cat-arrow_wf, 
cat-ob_wf, 
presheaf-term_wf, 
discrete-presheaf-type_wf, 
istype-universe
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
lambdaFormation_alt, 
sqequalHypSubstitution, 
setElimination, 
thin, 
rename, 
sqequalRule, 
extract_by_obid, 
dependent_functionElimination, 
Error :memTop, 
hypothesis, 
universeIsType, 
isectElimination, 
hypothesisEquality, 
applyEquality, 
inhabitedIsType, 
lambdaEquality_alt, 
axiomEquality, 
functionIsTypeImplies, 
isect_memberEquality_alt, 
isectIsTypeImplies, 
because_Cache, 
instantiate, 
universeEquality
Latex:
\mforall{}[C:SmallCategory].  \mforall{}[T:Type].  \mforall{}[X:ps\_context\{j:l\}(C)].  \mforall{}[t:\{X  \mvdash{}  \_:discr(T)\}].
    \mforall{}I,J:cat-ob(C).  \mforall{}f:cat-arrow(C)  J  I.  \mforall{}a:X(I).    (t(a)  =  t(f(a)))
Date html generated:
2020_05_20-PM-01_34_15
Last ObjectModification:
2020_04_02-PM-06_33_19
Theory : presheaf!models!of!type!theory
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