Nuprl Lemma : ps-sigma-unelim-elim-term
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[T:{X.Σ A B ⊢ _}]. ∀[t:{X.Σ A B ⊢ _:T}].
(((t)SigmaUnElim)SigmaElim = t ∈ {X.Σ A B ⊢ _:T})
Proof
Definitions occuring in Statement :
sigma-unelim-pscm: SigmaUnElim,
sigma-elim-pscm: SigmaElim,
presheaf-sigma: Σ A B,
psc-adjoin: X.A,
pscm-ap-term: (t)s,
presheaf-term: {X ⊢ _:A},
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
all: ∀x:A. B[x],
cat-comp: cat-comp(C),
compose: f o g,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
ps-sigma-elim-unelim,
presheaf-term_wf2,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-sigma_wf,
presheaf-type-cumulativity2,
presheaf-type_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
pscm-ap-type-is-id,
subtype_rel_self,
psc_map_wf,
pscm-id_wf,
iff_weakening_equal,
pscm-ap-term_wf,
pscm-ap-comp-term,
sigma-elim-pscm_wf,
sigma-unelim-pscm_wf,
pscm-ap-id-term
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
universeIsType,
instantiate,
applyEquality,
because_Cache,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
setElimination,
rename,
productElimination,
lambdaEquality_alt,
imageElimination,
universeEquality,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
hyp_replacement
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[T:\{X.\mSigma{} A B \mvdash{} \_\}].
\mforall{}[t:\{X.\mSigma{} A B \mvdash{} \_:T\}].
(((t)SigmaUnElim)SigmaElim = t)
Date html generated:
2020_05_20-PM-01_32_54
Last ObjectModification:
2020_04_03-AM-01_00_56
Theory : presheaf!models!of!type!theory
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