Nuprl Lemma : psc_map_cumulativity2
∀[C:SmallCategory]. ∀[X,Y,Z,U:ps_context{j:l}(C)].
(psc_map{j:l}(C; X; Z) ⊆r psc_map{j:l}(C; Y; U)) supposing
(sub_ps_context{j:l}(C; Y; X) and
sub_ps_context{j:l}(C; Z; U))
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,
member: t ∈ T,
subtype_rel: A ⊆r B,
istype: istype(T)
Lemmas referenced :
sub_ps_context_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
psc_map_wf,
psc_map_subtype3,
subtype_rel_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
independent_isectElimination,
lambdaEquality_alt
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y,Z,U:ps\_context\{j:l\}(C)].
(psc\_map\{j:l\}(C; X; Z) \msubseteq{}r psc\_map\{j:l\}(C; Y; U)) supposing
(sub\_ps\_context\{j:l\}(C; Y; X) and
sub\_ps\_context\{j:l\}(C; Z; U))
Date html generated:
2020_05_20-PM-01_25_12
Last ObjectModification:
2020_04_01-PM-00_17_51
Theory : presheaf!models!of!type!theory
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