Nuprl Lemma : psc_map_subtype
∀[C:SmallCategory]. ∀[X,Y,Z:ps_context{j:l}(C)].
psc_map{j:l}(C; Z; X) ⊆r psc_map{j:l}(C; Z; Y) supposing sub_ps_context{j:l}(C; X; 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],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
sub_ps_context: Y ⊆ X,
I_set: A(I),
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat,
ps_context: __⊢,
all: ∀x:A. B[x],
cat-ob: cat-ob(C),
pscm-id: 1(X),
pscm-comp: G o F,
compose: f o g
Lemmas referenced :
psc_map_wf,
sub_ps_context_wf,
pscm-equal,
pscm-comp_wf,
I_set_wf,
cat-ob_wf,
subtype_rel_self,
functor-ob_wf,
op-cat_wf,
small-category-cumulativity-2,
type-cat_wf,
subtype_rel-equal,
cat_ob_op_lemma,
subset-I_set,
cat_arrow_triple_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
sqequalHypSubstitution,
equalityElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
independent_isectElimination,
functionExtensionality,
setElimination,
rename,
functionEquality,
dependent_functionElimination,
universeEquality,
Error :memTop
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y,Z:ps\_context\{j:l\}(C)].
psc\_map\{j:l\}(C; Z; X) \msubseteq{}r psc\_map\{j:l\}(C; Z; Y) supposing sub\_ps\_context\{j:l\}(C; X; Y)
Date html generated:
2020_05_20-PM-01_25_04
Last ObjectModification:
2020_04_01-AM-11_51_06
Theory : presheaf!models!of!type!theory
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