Nuprl Lemma : pscm-subset-subtype
∀[C:SmallCategory]. ∀[A,B,Y,Z:ps_context{j:l}(C)].
(psc_map{j:l}(C; Y; A) ⊆r psc_map{j:l}(C; Z; B)) supposing
(sub_ps_context{j:l}(C; A; B) and
sub_ps_context{j:l}(C; Z; Y))
Proof
Definitions occuring in Statement :
sub_ps_context: Y ⊆ X,
psc_map: A ⟶ B,
ps_context: __⊢,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
guard: {T},
subtype_rel: A ⊆r B
Lemmas referenced :
psc_map_subtype3,
sub_ps_context_self,
subtype_rel_transitivity,
psc_map_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,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
independent_isectElimination,
hypothesis,
independent_pairFormation,
productElimination,
instantiate,
applyEquality,
sqequalRule,
universeIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B,Y,Z:ps\_context\{j:l\}(C)].
(psc\_map\{j:l\}(C; Y; A) \msubseteq{}r psc\_map\{j:l\}(C; Z; B)) supposing
(sub\_ps\_context\{j:l\}(C; A; B) and
sub\_ps\_context\{j:l\}(C; Z; Y))
Date html generated:
2020_05_20-PM-01_35_23
Last ObjectModification:
2020_04_02-PM-06_35_23
Theory : presheaf!models!of!type!theory
Home
Index