Nuprl Lemma : psc-restriction-comp
∀C:SmallCategory. ∀X:ps_context{j:l}(C). ∀I,J,K:cat-ob(C). ∀f:cat-arrow(C) J I. ∀g:cat-arrow(C) K J. ∀a:X(I).
(g(f(a)) = cat-comp(C) K J I g f(a) ∈ X(K))
Proof
Definitions occuring in Statement :
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
ps_context: __⊢,
cat-functor: Functor(C1;C2),
and: P ∧ Q,
psc-restriction: f(s),
pi2: snd(t),
small-category: SmallCategory,
spreadn: spread4,
cat-arrow: cat-arrow(C),
pi1: fst(t),
cat-ob: cat-ob(C),
type-cat: TypeCat,
cat-comp: cat-comp(C),
op-cat: op-cat(C),
cat-id: cat-id(C),
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
I_set: A(I),
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
compose: f o g
Lemmas referenced :
I_set_pair_redex_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
ob_pair_lemma,
subtype_rel_self,
iff_weakening_equal,
compose_wf,
I_set_wf,
cat-arrow_wf,
cat-ob_wf,
ps_context_wf,
small-category-cumulativity-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
introduction,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
applyEquality,
instantiate,
lambdaEquality_alt,
imageElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
inhabitedIsType
Latex:
\mforall{}C:SmallCategory. \mforall{}X:ps\_context\{j:l\}(C). \mforall{}I,J,K:cat-ob(C). \mforall{}f:cat-arrow(C) J I. \mforall{}g:cat-arrow(C) K J.
\mforall{}a:X(I).
(g(f(a)) = cat-comp(C) K J I g f(a))
Date html generated:
2020_05_20-PM-01_24_30
Last ObjectModification:
2020_04_01-AM-11_00_37
Theory : presheaf!models!of!type!theory
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