Nuprl Lemma : discrete-unary_wf
∀[C:SmallCategory]. ∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[X:ps_context{j:l}(C)]. ∀[t:{X ⊢ _:discr(A)}].
(discrete-unary(t;x.f[x]) ∈ {X ⊢ _:discr(B)})
Proof
Definitions occuring in Statement :
discrete-unary: discrete-unary(t;x.f[x]),
discrete-presheaf-type: discr(T),
presheaf-term: {X ⊢ _:A},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
discrete-unary: discrete-unary(t;x.f[x]),
presheaf-term: {X ⊢ _:A},
so_apply: x[s],
subtype_rel: A ⊆r B,
presheaf-type-at: A(a),
pi1: fst(t),
discrete-presheaf-type: discr(T),
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
presheaf-term-at_wf,
discrete-presheaf-type_wf,
subtype_rel_self,
subtype_rel-equal,
presheaf-type-at_wf,
I_set_wf,
cat-ob_wf,
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
discrete-presheaf-term-at-morph,
small-category-cumulativity-2,
ps_context_cumulativity2,
cat-arrow_wf,
psc-restriction_wf,
presheaf-type-ap-morph_wf,
presheaf-term_wf,
ps_context_wf,
istype-universe,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
universeIsType,
lambdaFormation_alt,
dependent_functionElimination,
Error :memTop,
instantiate,
cumulativity,
because_Cache,
inhabitedIsType,
functionIsType,
equalityIstype,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[t:\{X \mvdash{} \_:discr(A)\}].
(discrete-unary(t;x.f[x]) \mmember{} \{X \mvdash{} \_:discr(B)\})
Date html generated:
2020_05_20-PM-01_34_19
Last ObjectModification:
2020_04_02-PM-06_33_35
Theory : presheaf!models!of!type!theory
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