Nuprl Lemma : pscm-presheaf-id-fun
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[H:ps_context{j:l}(C)]. ∀[s:psc_map{j:l}(C; H; X)].
((presheaf-id-fun(X))s = presheaf-id-fun(H) ∈ {H ⊢ _:((A)s ⟶ (A)s)})
Proof
Definitions occuring in Statement :
presheaf-id-fun: presheaf-id-fun(X),
presheaf-fun: (A ⟶ B),
pscm-ap-term: (t)s,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
ps_context: __⊢,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
presheaf-type: {X ⊢ _},
psc-snd: q,
pscm+: tau+,
pscm-ap-term: (t)s,
psc-fst: p,
pscm-ap-type: (AF)s,
pscm-comp: G o F,
pscm-adjoin: (s;u),
pscm-ap: (s)x,
pi2: snd(t),
presheaf-id-fun: presheaf-id-fun(X),
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
psc_map_wf,
small-category-cumulativity-2,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
psc-snd_wf,
subtype_rel-equal,
presheaf-term_wf,
pscm-ap-type_wf,
presheaf-fun_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
pscm-presheaf-fun,
presheaf-type-cumulativity2,
ps_context_cumulativity2,
subtype_rel_self,
iff_weakening_equal,
equal_functionality_wrt_subtype_rel2,
pscm-presheaf-lam
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
universeIsType,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
setElimination,
rename,
productElimination,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[H:ps\_context\{j:l\}(C)].
\mforall{}[s:psc\_map\{j:l\}(C; H; X)].
((presheaf-id-fun(X))s = presheaf-id-fun(H))
Date html generated:
2020_05_20-PM-01_30_33
Last ObjectModification:
2020_04_02-PM-05_58_05
Theory : presheaf!models!of!type!theory
Home
Index