Nuprl Lemma : pscm-subset-codomain
∀[C:SmallCategory]. ∀[X,Y,Z:ps_context{j:l}(C)].
psc_map{j:l}(C; X; Y) ⊆r psc_map{j:l}(C; X; Z) supposing sub_ps_context{j:l}(C; Y; Z)
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,
subtype_rel: A ⊆r B,
member: t ∈ T,
sub_ps_context: Y ⊆ X,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
all: ∀x:A. B[x],
ps_context: __⊢,
type-cat: TypeCat,
cat-arrow: cat-arrow(C),
pi2: snd(t),
pi1: fst(t),
cat-ob: cat-ob(C),
pscm-id: 1(X),
pscm-comp: G o F,
compose: f o g,
squash: ↓T,
guard: {T}
Lemmas referenced :
pscm-comp_wf,
psc_map_wf,
sub_ps_context_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
cat-ob_wf,
op-cat_wf,
cat-arrow_wf,
type-cat_wf,
functor-ob_wf,
cat-comp_wf,
functor-arrow_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
dependent_set_memberEquality_alt,
functionIsType,
equalityIstype,
functionExtensionality,
universeEquality,
setElimination,
rename,
inhabitedIsType,
functionEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaFormation_alt,
dependent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y,Z:ps\_context\{j:l\}(C)].
psc\_map\{j:l\}(C; X; Y) \msubseteq{}r psc\_map\{j:l\}(C; X; Z) supposing sub\_ps\_context\{j:l\}(C; Y; Z)
Date html generated:
2020_05_20-PM-01_35_14
Last ObjectModification:
2020_04_02-PM-06_35_07
Theory : presheaf!models!of!type!theory
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