Nuprl Lemma : pscm-ap-term_wf
∀[C:SmallCategory]. ∀[Delta,Gamma:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢ _}]. ∀[s:psc_map{j:l}(C; Delta; Gamma)].
∀[t:{Gamma ⊢ _:A}].
((t)s ∈ {Delta ⊢ _:(A)s})
Proof
Definitions occuring in Statement :
pscm-ap-term: (t)s,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
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-type: {X ⊢ _},
presheaf-term: {X ⊢ _:A},
all: ∀x:A. B[x],
pscm-ap-term: (t)s,
pscm-ap-type: (AF)s,
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
presheaf-term_wf,
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
pscm-ap_wf,
I_set_wf,
cat-ob_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
psc-restriction_wf,
subtype_rel-equal,
ps_context_cumulativity2,
small-category-cumulativity-2,
pscm-ap-restriction,
subtype_rel_self,
iff_weakening_equal,
cat-arrow_wf,
psc_map_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
productElimination,
sqequalRule,
dependent_functionElimination,
Error :memTop,
hypothesis,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
applyEquality,
universeIsType,
lambdaFormation_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
because_Cache,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
inhabitedIsType,
functionIsType,
equalityIstype
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Delta,Gamma:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[s:psc\_map\{j:l\}(C;
Delta;
Gamma)].
\mforall{}[t:\{Gamma \mvdash{} \_:A\}].
((t)s \mmember{} \{Delta \mvdash{} \_:(A)s\})
Date html generated:
2020_05_20-PM-01_26_54
Last ObjectModification:
2020_04_01-PM-00_17_24
Theory : presheaf!models!of!type!theory
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