Nuprl Lemma : presheaf-fun-as-presheaf-pi
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A,B:{X ⊢ _}]. ((X ⊢ A ⟶ B) = X ⊢ ΠA (B)p ∈ {X ⊢ _})
Proof
Definitions occuring in Statement :
presheaf-fun: (A ⟶ B),
presheaf-pi: ΠA B,
psc-fst: p,
pscm-ap-type: (AF)s,
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,
uimplies: b supposing a,
presheaf-pi: ΠA B,
presheaf-fun: (A ⟶ B),
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
pscm-ap: (s)x,
pscm-id: 1(X),
psc-restriction: f(s),
pi2: snd(t)
Lemmas referenced :
presheaf-type-equal2,
presheaf-fun_wf,
presheaf-pi_wf,
pscm-ap-type_wf,
ps_context_cumulativity2,
small-category-cumulativity-2,
psc-adjoin_wf,
presheaf-type-cumulativity2,
psc-fst_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
cat-ob_wf,
cat-arrow_wf,
I_set_wf,
psc-restriction_wf,
pscm-ap-type-at,
cc_fst_adjoin_set_lemma,
presheaf-type-at_wf,
equal_wf,
psc-restriction-comp,
pscm-ap_wf,
psc-adjoin-set_wf,
subtype_rel_self,
iff_weakening_equal,
pscm-presheaf-type-ap-morph,
presheaf-type-ap-morph_wf,
cat-comp_wf,
subtype_rel-equal,
psc-adjoin-set-restriction,
squash_wf,
true_wf,
istype-universe,
presheaf-fun-family_wf,
pscm-id_wf,
presheaf-fun-family-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
universeIsType,
dependent_pairEquality_alt,
functionIsType,
functionExtensionality,
setEquality,
functionEquality,
Error :memTop,
dependent_functionElimination,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
natural_numberEquality,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A,B:\{X \mvdash{} \_\}]. ((X \mvdash{} A {}\mrightarrow{} B) = X \mvdash{} \mPi{}A (B)p)
Date html generated:
2020_05_20-PM-01_30_06
Last ObjectModification:
2020_04_02-PM-06_00_13
Theory : presheaf!models!of!type!theory
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