Nuprl Lemma : ps-subset-restriction
∀[C:SmallCategory]. ∀[X,Y:ps_context{j:l}(C)].
∀[I,J:cat-ob(C)]. ∀[f:cat-arrow(C) J I]. ∀[a:Y(I)]. (f(a) = f(a) ∈ X(J)) supposing sub_ps_context{j:l}(C; Y; X)
Proof
Definitions occuring in Statement :
sub_ps_context: Y ⊆ X,
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-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sub_ps_context: Y ⊆ X,
member: t ∈ T,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
squash: ↓T,
small-category: SmallCategory,
spreadn: spread4,
and: P ∧ Q,
cat-arrow: cat-arrow(C),
pi2: snd(t),
pi1: fst(t),
cat-ob: cat-ob(C),
functor-arrow: arrow(F),
type-cat: TypeCat,
cat-comp: cat-comp(C),
op-cat: op-cat(C),
pscm-id: 1(X),
compose: f o g,
psc-restriction: f(s),
I_set: A(I),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x]
Lemmas referenced :
I_set_wf,
cat-arrow_wf,
cat-ob_wf,
sub_ps_context_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
cut,
applyLambdaEquality,
setElimination,
thin,
rename,
hypothesis,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
introduction,
imageElimination,
productElimination,
universeIsType,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
instantiate,
dependent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y:ps\_context\{j:l\}(C)].
\mforall{}[I,J:cat-ob(C)]. \mforall{}[f:cat-arrow(C) J I]. \mforall{}[a:Y(I)]. (f(a) = f(a))
supposing sub\_ps\_context\{j:l\}(C; Y; X)
Date html generated:
2020_05_20-PM-01_24_57
Last ObjectModification:
2020_04_01-AM-11_00_45
Theory : presheaf!models!of!type!theory
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