Nuprl Lemma : pscm-adjoin-fst-snd
∀[C:SmallCategory]. ∀[Gamma:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢ _}].
((p;q) = 1(Gamma.A) ∈ psc_map{[i | j]:l}(C; Gamma.A; Gamma.A))
Proof
Definitions occuring in Statement :
pscm-adjoin: (s;u),
psc-snd: q,
psc-fst: p,
psc-adjoin: X.A,
presheaf-type: {X ⊢ _},
pscm-id: 1(X),
psc_map: A ⟶ B,
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-type: {X ⊢ _},
psc-snd: q,
psc-fst: p,
pscm-adjoin: (s;u),
psc-adjoin: X.A,
I_set: A(I),
all: ∀x:A. B[x],
pscm-ap: (s)x,
functor-ob: ob(F),
pi1: fst(t),
pi2: snd(t),
pscm-id: 1(X),
uimplies: b supposing a
Lemmas referenced :
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-id_wf,
ob_pair_lemma,
presheaf_type_at_pair_lemma,
I_set_wf,
cat-ob_wf,
pscm-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
because_Cache,
setElimination,
rename,
productElimination,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
dependent_pairEquality_alt,
productIsType,
functionExtensionality_alt,
independent_isectElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. ((p;q) = 1(Gamma.A))
Date html generated:
2020_05_20-PM-01_28_16
Last ObjectModification:
2020_04_02-PM-02_58_58
Theory : presheaf!models!of!type!theory
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