Nuprl Lemma : sigma-elim-pscm_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}].
(SigmaElim ∈ psc_map{[i | j]:l}(C; X.Σ A B; X.A.B))
Proof
Definitions occuring in Statement :
sigma-elim-pscm: SigmaElim,
presheaf-sigma: Σ A B,
psc-adjoin: X.A,
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
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,
subtype_rel: A ⊆r B,
presheaf-sigma: Σ A B,
psc-adjoin: X.A,
all: ∀x:A. B[x],
sigma-elim-pscm: SigmaElim,
spreadn: spread3,
psc-restriction: f(s),
pi2: snd(t),
presheaf-type-ap-morph: (u a f),
pi1: fst(t),
psc-adjoin-set: (v;u)
Lemmas referenced :
psc-map-is,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-sigma_wf,
presheaf-type-cumulativity2,
I_set_pair_redex_lemma,
presheaf_type_at_pair_lemma,
psc-adjoin-set_wf,
I_set_wf,
cat-ob_wf,
psc-adjoin-set-restriction,
psc-restriction_wf,
presheaf-type-ap-morph_wf,
subtype_rel_self,
presheaf-type-at_wf,
small-category-cumulativity-2,
cat-arrow_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
dependent_set_memberEquality_alt,
functionExtensionality,
dependent_functionElimination,
Error :memTop,
productElimination,
lambdaFormation_alt,
universeIsType,
inhabitedIsType,
functionIsType,
equalityIstype,
lambdaEquality_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}].
(SigmaElim \mmember{} psc\_map\{[i | j]:l\}(C; X.\mSigma{} A B; X.A.B))
Date html generated:
2020_05_20-PM-01_32_34
Last ObjectModification:
2020_04_02-PM-06_41_28
Theory : presheaf!models!of!type!theory
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