Nuprl Lemma : subset-I_set
∀[C:SmallCategory]. ∀[X,Y:ps_context{j:l}(C)]. ∀I:cat-ob(C). (Y(I) ⊆r X(I)) supposing sub_ps_context{j:l}(C; Y; X)
Proof
Definitions occuring in Statement :
sub_ps_context: Y ⊆ X,
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
sub_ps_context: Y ⊆ X,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
squash: ↓T,
small-category: SmallCategory,
spreadn: spread4,
and: P ∧ Q,
pscm-id: 1(X),
type-cat: TypeCat,
cat-arrow: cat-arrow(C),
op-cat: op-cat(C),
cat-ob: cat-ob(C),
pi2: snd(t),
pi1: fst(t),
I_set: A(I)
Lemmas referenced :
sub_ps_context_wf,
cat_ob_pair_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
applyLambdaEquality,
setElimination,
thin,
rename,
hypothesis,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
imageElimination,
because_Cache,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
extract_by_obid,
isectElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
productElimination,
applyEquality,
Error :memTop,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y:ps\_context\{j:l\}(C)].
\mforall{}I:cat-ob(C). (Y(I) \msubseteq{}r X(I)) supposing sub\_ps\_context\{j:l\}(C; Y; X)
Date html generated:
2020_05_20-PM-01_24_54
Last ObjectModification:
2020_04_01-AM-11_00_44
Theory : presheaf!models!of!type!theory
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