Nuprl Lemma : presheaf-snd_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ A B}]. (p.2 ∈ {X ⊢ _:(B)[p.1]})
Proof
Definitions occuring in Statement :
presheaf-snd: p.2,
presheaf-fst: p.1,
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-snd: p.2,
subtype_rel: A ⊆r B,
presheaf-term: {X ⊢ _:A},
presheaf-sigma: Σ A B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pi1: fst(t),
pi2: snd(t),
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
pscm-id-adjoin: [u],
pscm-ap-type: (AF)s,
pscm-adjoin: (s;u),
pscm-ap: (s)x,
psc-adjoin-set: (v;u),
presheaf-fst: p.1,
pscm-id: 1(X),
and: P ∧ Q,
presheaf-type-ap-morph: (u a f),
psc-adjoin: X.A,
I_set: A(I),
functor-ob: ob(F),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
squash: ↓T,
true: True,
guard: {T}
Lemmas referenced :
presheaf-fst_wf,
presheaf-term_wf,
presheaf-sigma_wf,
presheaf-type_wf,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
ps_context_wf,
small-category_wf,
presheaf_type_at_pair_lemma,
I_set_wf,
cat-ob_wf,
subtype_rel_self,
presheaf-type-at_wf,
psc-adjoin-set_wf,
pscm-ap-type-at,
cat-arrow_wf,
presheaf_type_ap_morph_pair_lemma,
ob_pair_lemma,
psc-restriction_wf,
pi1_wf_top,
subtype_rel_product,
top_wf,
equal_wf,
subtype_rel_weakening,
ext-eq_weakening,
pscm-ap-type_wf,
pscm-id-adjoin_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
lambdaFormation_alt,
productElimination,
equalityIstype,
independent_functionElimination,
promote_hyp,
dependent_pairEquality_alt,
applyLambdaEquality,
independent_pairEquality,
because_Cache,
independent_isectElimination,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
functionIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[p:\{X \mvdash{} \_:\mSigma{} A B\}].
(p.2 \mmember{} \{X \mvdash{} \_:(B)[p.1]\})
Date html generated:
2020_05_20-PM-01_32_05
Last ObjectModification:
2020_04_02-PM-06_32_45
Theory : presheaf!models!of!type!theory
Home
Index