Nuprl Lemma : presheaf-sigma-equal
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w,y:{X ⊢ _:Σ A B}].
(w = y ∈ {X ⊢ _:Σ A B}) supposing ((w.2 = y.2 ∈ {X ⊢ _:(B)[w.1]}) and (w.1 = y.1 ∈ {X ⊢ _:A}))
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: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
and: P ∧ Q
Lemmas referenced :
presheaf-pair-eta,
presheaf-pair_wf,
presheaf-term_wf,
pscm-ap-type_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-id-adjoin_wf,
small-category-cumulativity-2,
presheaf-fst_wf,
presheaf-snd_wf,
subtype_rel-equal,
presheaf-sigma_wf,
presheaf-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityIstype,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w,y:\{X \mvdash{} \_:\mSigma{} A B\}].
(w = y) supposing ((w.2 = y.2) and (w.1 = y.1))
Date html generated:
2020_05_20-PM-01_33_37
Last ObjectModification:
2020_04_03-AM-01_04_48
Theory : presheaf!models!of!type!theory
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