Nuprl Lemma : psc-restriction_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[I,J:cat-ob(C)]. ∀[f:cat-arrow(C) J I]. ∀[s:X(I)]. (f(s) ∈ X(J))
Proof
Definitions occuring in Statement :
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
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,
psc-restriction: f(s),
ps_context: __⊢,
cat-functor: Functor(C1;C2),
and: P ∧ Q,
all: ∀x:A. B[x],
pi2: snd(t),
I_set: A(I),
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
I_set_pair_redex_lemma,
ob_pair_lemma,
subtype_rel-equal,
cat-ob_wf,
op-cat_wf,
cat_ob_op_lemma,
cat-arrow_wf,
op-cat-arrow,
I_set_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
because_Cache,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[I,J:cat-ob(C)]. \mforall{}[f:cat-arrow(C) J I]. \mforall{}[s:X(I)].
(f(s) \mmember{} X(J))
Date html generated:
2020_05_20-PM-01_23_15
Last ObjectModification:
2020_04_02-AM-11_20_36
Theory : presheaf!models!of!type!theory
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