Nuprl Lemma : psc-restriction-id
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[I:cat-ob(C)]. ∀[s:X(I)]. (cat-id(C) I(s) = s ∈ X(I))
Proof
Definitions occuring in Statement :
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T,
cat-id: cat-id(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ps_context: __⊢,
cat-functor: Functor(C1;C2),
and: P ∧ Q,
psc-restriction: f(s),
all: ∀x:A. B[x],
pi2: snd(t),
small-category: SmallCategory,
spreadn: spread4,
type-cat: TypeCat,
op-cat: op-cat(C),
cat-id: cat-id(C),
pi1: fst(t),
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
I_set_pair_redex_lemma,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cat_ob_pair_lemma,
I_set_wf,
cat-ob_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
universeIsType,
isectElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
natural_numberEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[I:cat-ob(C)]. \mforall{}[s:X(I)]. (cat-id(C) I(s) = s)
Date html generated:
2020_05_20-PM-01_24_24
Last ObjectModification:
2020_04_01-AM-11_00_34
Theory : presheaf!models!of!type!theory
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