Nuprl Lemma : presheaf-pair-eta
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:Σ A B}].
(presheaf-pair(w.1;w.2) = w ∈ {X ⊢ _:Σ A B})
Proof
Definitions occuring in Statement :
presheaf-pair: presheaf-pair(u;v),
presheaf-snd: p.2,
presheaf-fst: p.1,
presheaf-sigma: Σ A B,
psc-adjoin: X.A,
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,
subtype_rel: A ⊆r B,
presheaf-term: {X ⊢ _:A},
presheaf-pair: presheaf-pair(u;v),
presheaf-sigma: Σ A B,
all: ∀x:A. B[x],
presheaf-snd: p.2,
presheaf-fst: p.1,
implies: P ⇒ Q,
pi1: fst(t),
pi2: snd(t),
uimplies: b supposing a,
and: P ∧ Q
Lemmas referenced :
presheaf-term-equal,
presheaf-sigma_wf,
presheaf-pair_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
psc-adjoin_wf,
presheaf-fst_wf,
presheaf-snd_wf,
presheaf_type_at_pair_lemma,
subtype_rel-equal,
presheaf-type-at_wf,
psc-adjoin-set_wf,
I_set_wf,
cat-ob_wf,
presheaf-term_wf,
presheaf-type_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
functionExtensionality_alt,
dependent_functionElimination,
Error :memTop,
independent_pairEquality,
inhabitedIsType,
lambdaFormation_alt,
productElimination,
equalityIstype,
equalityTransitivity,
independent_functionElimination,
dependent_pairEquality_alt,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
applyLambdaEquality,
universeIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:\mSigma{} A B\}].
(presheaf-pair(w.1;w.2) = w)
Date html generated:
2020_05_20-PM-01_33_33
Last ObjectModification:
2020_04_02-PM-06_31_28
Theory : presheaf!models!of!type!theory
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