Nuprl Lemma : ps-context-map-comp2
∀[C:SmallCategory]. ∀[G:ps_context{j:l}(C)]. ∀[I,J:cat-ob(C)]. ∀[f:cat-arrow(C) J I]. ∀[a:G(I)].
(<a> o <f> = <f(a)> ∈ psc_map{[i | j]:l}(C; Yoneda(J); G))
Proof
Definitions occuring in Statement :
pscm-comp: G o F,
ps-context-map: <rho>,
psc_map: A ⟶ B,
Yoneda: Yoneda(I),
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uall: ∀[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],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
I_set: A(I),
functor-ob: ob(F),
Yoneda: Yoneda(I),
uimplies: b supposing a,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
type-cat: TypeCat,
pscm-comp: G o F,
compose: f o g,
ps-context-map: <rho>,
functor-arrow: arrow(F),
psc-restriction: f(s)
Lemmas referenced :
pscm-equal,
Yoneda_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
pscm-comp_wf,
ps-context-map_wf,
subtype_rel_self,
I_set_wf,
cat-arrow_wf,
cat-ob_wf,
ps_context_wf,
small-category_wf,
psc-restriction_wf,
subtype_rel-equal,
op-cat_wf,
cat_ob_op_lemma,
psc-restriction-comp,
arrow_pair_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
dependent_functionElimination,
functionExtensionality,
independent_isectElimination,
universeIsType,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
Error :memTop
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[G:ps\_context\{j:l\}(C)]. \mforall{}[I,J:cat-ob(C)]. \mforall{}[f:cat-arrow(C) J I]. \mforall{}[a:G(I)].
(<a> o <f> = <f(a)>)
Date html generated:
2020_05_20-PM-01_27_04
Last ObjectModification:
2020_04_03-PM-00_28_31
Theory : presheaf!models!of!type!theory
Home
Index