Nuprl Lemma : presheaf-lambda_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ((λb) ∈ {X ⊢ _:ΠA B})
Proof
Definitions occuring in Statement :
presheaf-lambda: (λb),
presheaf-pi: ΠA B,
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-term: {X ⊢ _:A},
subtype_rel: A ⊆r B,
presheaf-lambda: (λb),
presheaf-pi: ΠA B,
all: ∀x:A. B[x],
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
uimplies: b supposing a,
squash: ↓T,
true: True,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
presheaf-type: {X ⊢ _},
presheaf-type-ap-morph: (u a f),
psc-adjoin-set: (v;u),
psc-restriction: f(s),
pi2: snd(t),
psc-adjoin: X.A,
pi1: fst(t)
Lemmas referenced :
presheaf-term_wf,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
presheaf_type_at_pair_lemma,
psc-adjoin-set_wf,
psc-restriction_wf,
presheaf-type-at_wf,
cat-arrow_wf,
presheaf-type-ap-morph_wf,
cat-comp_wf,
subtype_rel-equal,
psc-restriction-comp,
psc-adjoin-set-restriction,
equal_wf,
squash_wf,
true_wf,
istype-universe,
I_set_wf,
subtype_rel_self,
iff_weakening_equal,
cat-ob_wf,
presheaf_type_ap_morph_pair_lemma,
presheaf-pi_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
setElimination,
rename,
lambdaFormation_alt,
functionIsType,
equalityIstype,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
hyp_replacement,
functionExtensionality_alt,
independent_pairFormation,
productIsType,
applyLambdaEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[b:\{X.A \mvdash{} \_:B\}].
((\mlambda{}b) \mmember{} \{X \mvdash{} \_:\mPi{}A B\})
Date html generated:
2020_05_20-PM-01_30_14
Last ObjectModification:
2020_04_02-PM-03_06_47
Theory : presheaf!models!of!type!theory
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