Nuprl Lemma : pscm-ap_wf
∀[C:SmallCategory]. ∀[X,Y:ps_context{j:l}(C)]. ∀[s:psc_map{j:l}(C; X; Y)]. ∀[I:cat-ob(C)]. ∀[x:X(I)]. ((s)x ∈ Y(I))
Proof
Definitions occuring in Statement :
pscm-ap: (s)x,
psc_map: A ⟶ B,
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pscm-ap: (s)x,
ps_context: __⊢,
cat-functor: Functor(C1;C2),
all: ∀x:A. B[x],
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
small-category: SmallCategory,
spreadn: spread4,
and: P ∧ Q,
functor-ob: ob(F),
type-cat: TypeCat,
cat-arrow: cat-arrow(C),
op-cat: op-cat(C),
cat-ob: cat-ob(C),
pi2: snd(t),
pi1: fst(t),
subtype_rel: A ⊆r B
Lemmas referenced :
I_set_pair_redex_lemma,
cat_ob_pair_lemma,
I_set_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,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
sqequalRule,
applyEquality,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isectElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
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:cat-ob(C)].
\mforall{}[x:X(I)].
((s)x \mmember{} Y(I))
Date html generated:
2020_05_20-PM-01_24_03
Last ObjectModification:
2020_04_01-AM-10_46_57
Theory : presheaf!models!of!type!theory
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