Nuprl Lemma : presheaf-type-ap-morph-id
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[I:cat-ob(C)]. ∀[f:cat-arrow(C) I I]. ∀[a:X(I)]. ∀[u:A(a)].
(u a f) = u ∈ A(a) supposing f = (cat-id(C) I) ∈ (cat-arrow(C) I I)
Proof
Definitions occuring in Statement :
presheaf-type-ap-morph: (u a f),
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T,
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
presheaf-type: {X ⊢ _},
presheaf-type-ap-morph: (u a f),
all: ∀x:A. B[x],
pi2: snd(t)
Lemmas referenced :
equal_wf,
presheaf-type-at_wf,
presheaf-type-ap-morph_wf,
subtype_rel-equal,
psc-restriction_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
squash_wf,
true_wf,
istype-universe,
I_set_wf,
psc-restriction-when-id,
subtype_rel_self,
iff_weakening_equal,
cat-id_wf,
cat-arrow_wf,
cat-ob_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
presheaf_type_at_pair_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
hypothesis,
because_Cache,
equalityIstype,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
thin,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
instantiate,
sqequalRule,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
universeIsType,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
natural_numberEquality,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
dependent_functionElimination,
Error :memTop
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[I:cat-ob(C)]. \mforall{}[f:cat-arrow(C) I I].
\mforall{}[a:X(I)]. \mforall{}[u:A(a)].
(u a f) = u supposing f = (cat-id(C) I)
Date html generated:
2020_05_20-PM-01_26_03
Last ObjectModification:
2020_04_01-PM-00_01_24
Theory : presheaf!models!of!type!theory
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