Nuprl Lemma : pscm-ap-restriction
∀C:SmallCategory. ∀X,Y:ps_context{j:l}(C). ∀s:psc_map{j:l}(C; X; Y). ∀I,J:cat-ob(C). ∀f:cat-arrow(C) J I. ∀a:X(I).
(f((s)a) = (s)f(a) ∈ Y(J))
Proof
Definitions occuring in Statement :
pscm-ap: (s)x,
psc_map: A ⟶ B,
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
all: ∀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 :
all: ∀x:A. B[x],
ps_context: __⊢,
cat-functor: Functor(C1;C2),
and: P ∧ Q,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
small-category: SmallCategory,
spreadn: spread4,
I_set: A(I),
member: t ∈ T,
cat-arrow: cat-arrow(C),
pi2: snd(t),
pi1: fst(t),
cat-ob: cat-ob(C),
functor-arrow: arrow(F),
functor-ob: ob(F),
type-cat: TypeCat,
cat-comp: cat-comp(C),
op-cat: op-cat(C),
compose: f o g,
psc-restriction: f(s),
pscm-ap: (s)x,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
ob_pair_lemma,
cat_id_tuple_lemma,
I_set_wf,
cat-arrow_wf,
cat-ob_wf,
psc_map_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
introduction,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
hypothesisEquality,
applyLambdaEquality,
applyEquality,
universeIsType,
isectElimination,
inhabitedIsType,
instantiate,
because_Cache
Latex:
\mforall{}C:SmallCategory. \mforall{}X,Y:ps\_context\{j:l\}(C). \mforall{}s:psc\_map\{j:l\}(C; X; Y). \mforall{}I,J:cat-ob(C).
\mforall{}f:cat-arrow(C) J I. \mforall{}a:X(I).
(f((s)a) = (s)f(a))
Date html generated:
2020_05_20-PM-01_24_38
Last ObjectModification:
2020_04_01-AM-11_00_40
Theory : presheaf!models!of!type!theory
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