Nuprl Lemma : psc-restriction-when-id
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[I:cat-ob(C)]. ∀[s:X(I)]. ∀[f:cat-arrow(C) I I].
f(s) = s ∈ X(I) supposing f = (cat-id(C) I) ∈ (cat-arrow(C) I I)
Proof
Definitions occuring in Statement :
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T,
cat-id: cat-id(C),
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,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
psc-restriction-id,
small-category-cumulativity-2,
ps_context_cumulativity2,
equal_wf,
I_set_wf,
psc-restriction_wf,
cat-id_wf,
cat-arrow_wf,
cat-ob_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
because_Cache,
equalityIstype,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[I:cat-ob(C)]. \mforall{}[s:X(I)]. \mforall{}[f:cat-arrow(C) I I].
f(s) = s supposing f = (cat-id(C) I)
Date html generated:
2020_05_20-PM-01_24_27
Last ObjectModification:
2020_04_01-AM-11_00_36
Theory : presheaf!models!of!type!theory
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