Nuprl Lemma : presheaf-pair_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:{X ⊢ _:(B)[u]}].
(presheaf-pair(u;v) ∈ {X ⊢ _:Σ A B})
Proof
Definitions occuring in Statement :
presheaf-pair: presheaf-pair(u;v),
presheaf-sigma: Σ A B,
pscm-id-adjoin: [u],
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-pair: presheaf-pair(u;v),
subtype_rel: A ⊆r B,
presheaf-term: {X ⊢ _:A},
presheaf-sigma: Σ A B,
all: ∀x:A. B[x],
uimplies: b supposing a,
squash: ↓T,
true: True,
pi1: fst(t),
pi2: snd(t),
presheaf-type: {X ⊢ _},
presheaf-type-ap-morph: (u a f),
presheaf-type-at: A(a),
and: P ∧ Q,
guard: {T},
pscm-id-adjoin: [u],
pscm-ap-type: (AF)s,
pscm-id: 1(X),
pscm-adjoin: (s;u),
pscm-ap: (s)x,
psc-adjoin-set: (v;u),
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
psc-adjoin: X.A,
I_set: A(I),
functor-ob: ob(F),
ps_context: __⊢,
cat-functor: Functor(C1;C2),
cat-ob: cat-ob(C),
type-cat: TypeCat,
istype: istype(T),
label: ...$L... t,
psc-restriction: f(s)
Lemmas referenced :
presheaf-term_wf,
pscm-ap-type_wf,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-id-adjoin_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
presheaf_type_at_pair_lemma,
subtype_rel-equal,
presheaf-type-at_wf,
psc-adjoin-set_wf,
pscm-ap-type-at,
pscm-id-adjoin-ap,
I_set_wf,
cat-ob_wf,
presheaf_type_ap_morph_pair_lemma,
psc-restriction_wf,
cat-arrow_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
pscm-adjoin-ap,
trivial-equal,
ob_pair_lemma,
op-cat_wf,
cat_ob_op_lemma,
pscm-ap_wf,
pscm-ap-restriction,
presheaf-sigma_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
because_Cache,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
dependent_pairEquality_alt,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
promote_hyp,
lambdaFormation_alt,
productElimination,
universeEquality,
independent_functionElimination,
productEquality,
cumulativity,
hyp_replacement,
equalityIstype,
functionEquality,
functionIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}].
\mforall{}[v:\{X \mvdash{} \_:(B)[u]\}].
(presheaf-pair(u;v) \mmember{} \{X \mvdash{} \_:\mSigma{} A B\})
Date html generated:
2020_05_20-PM-01_32_31
Last ObjectModification:
2020_04_03-AM-10_22_38
Theory : presheaf!models!of!type!theory
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