Nuprl Lemma : implies-sub_ps_context
∀[C:SmallCategory]. ∀[Y,X:ps_context{j:l}(C)].
(sub_ps_context{j:l}(C; Y; X)) supposing
((∀A,B:cat-ob(C). ∀g:cat-arrow(C) B A. ∀rho:Y(A). (g(rho) = g(rho) ∈ X(B))) and
(∀I:cat-ob(C). (Y(I) ⊆r X(I))))
Proof
Definitions occuring in Statement :
sub_ps_context: Y ⊆ X,
psc-restriction: f(s),
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],
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,
pscm-id: 1(X),
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
small-category: SmallCategory,
spreadn: spread4,
and: P ∧ Q,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat,
op-cat: op-cat(C),
functor-ob: ob(F),
I_set: A(I),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
functor-arrow: arrow(F),
compose: f o g,
psc-restriction: f(s),
ps_context: __⊢,
guard: {T}
Lemmas referenced :
I_set_wf,
cat_ob_pair_lemma,
cat_comp_tuple_lemma,
cat-ob_wf,
op-cat_wf,
cat-arrow_wf,
type-cat_wf,
functor-ob_wf,
cat-comp_wf,
small-category-cumulativity-2,
functor-arrow_wf,
psc-restriction_wf,
ps_context_cumulativity2,
subtype_rel_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
dependent_set_memberEquality_alt,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
hypothesis,
dependent_functionElimination,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
equalityTransitivity,
equalitySymmetry,
Error :memTop,
because_Cache,
lambdaFormation_alt,
functionExtensionality_alt,
functionIsType,
equalityIstype,
instantiate
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Y,X:ps\_context\{j:l\}(C)].
(sub\_ps\_context\{j:l\}(C; Y; X)) supposing
((\mforall{}A,B:cat-ob(C). \mforall{}g:cat-arrow(C) B A. \mforall{}rho:Y(A). (g(rho) = g(rho))) and
(\mforall{}I:cat-ob(C). (Y(I) \msubseteq{}r X(I))))
Date html generated:
2020_05_20-PM-01_24_51
Last ObjectModification:
2020_04_01-PM-01_57_34
Theory : presheaf!models!of!type!theory
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