Nuprl Lemma : presheaf-sigma_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. (Σ A B ∈ X ⊢ )
Proof
Definitions occuring in Statement :
presheaf-sigma: Σ A B,
psc-adjoin: X.A,
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-sigma: Σ A B,
presheaf-type: {X ⊢ _},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
psc-restriction: f(s),
pi2: snd(t),
psc-adjoin: X.A,
psc-adjoin-set: (v;u),
pi1: fst(t),
and: P ∧ Q,
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
presheaf-type-at_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
psc-adjoin-set_wf,
I_set_wf,
cat-ob_wf,
presheaf-type-ap-morph_wf,
pi1_wf_top,
pi2_wf,
subtype_rel_self,
small-category-cumulativity-2,
psc-restriction_wf,
cat-arrow_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cat-comp_wf,
presheaf-type-ap-morph-comp,
iff_weakening_equal,
presheaf-type-ap-morph-comp-eq,
psc-adjoin-set-restriction,
cat-id_wf,
subtype_rel-equal,
psc-restriction-id,
psc-restriction-comp,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
presheaf-type-ap-morph-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
lambdaEquality_alt,
productEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
dependent_functionElimination,
universeIsType,
productElimination,
independent_pairEquality,
Error :memTop,
productIsType,
inhabitedIsType,
functionIsType,
lambdaFormation_alt,
independent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
equalityIstype,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. (\mSigma{} A B \mmember{} X \mvdash{} )
Date html generated:
2020_05_20-PM-01_31_23
Last ObjectModification:
2020_04_02-PM-03_03_10
Theory : presheaf!models!of!type!theory
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