Nuprl Lemma : pscm-presheaf-pi-family
∀C:SmallCategory. ∀X,Delta:ps_context{j:l}(C). ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:psc_map{j:l}(C; Delta; X). ∀I:cat-ob(C).
∀a:Delta(I).
(presheaf-pi-family(C; X; A; B; I; (s)a) = presheaf-pi-family(C; Delta; (A)s; (B)(s o p;q); I; a) ∈ Type)
Proof
Definitions occuring in Statement :
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
pscm-adjoin: (s;u),
psc-snd: q,
psc-fst: p,
psc-adjoin: X.A,
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
pscm-comp: G o F,
pscm-ap: (s)x,
psc_map: A ⟶ B,
I_set: A(I),
ps_context: __⊢,
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
cat-comp: cat-comp(C),
compose: f o g,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
squash: ↓T,
true: True,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
presheaf-type-ap-morph: (u a f),
presheaf-type: {X ⊢ _},
pscm-ap-type: (AF)s,
presheaf-type-at: A(a)
Lemmas referenced :
pscm-comp_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
pscm-ap-type_wf,
presheaf-type-cumulativity2,
psc-fst_wf,
subtype_rel_self,
psc_map_wf,
small-category-cumulativity-2,
psc-snd_wf,
cat-arrow_wf,
I_set_wf,
cat-ob_wf,
presheaf-type_wf,
ps_context_wf,
pscm-ap-type-at,
presheaf-type-at_wf,
pscm-ap-restriction,
squash_wf,
true_wf,
small-category_wf,
pscm-adjoin-ap,
csm_comp_fst_adjoin_set_lemma,
cc_snd_adjoin_set_lemma,
psc-adjoin-set_wf,
equal_wf,
istype-universe,
psc-adjoin-set-restriction,
psc-restriction_wf,
pscm-ap_wf,
iff_weakening_equal,
presheaf-type-ap-morph_wf,
subtype_rel-equal,
pscm-presheaf-type-ap-morph,
presheaf_type_at_pair_lemma,
cat-comp_wf,
psc-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
setEquality,
functionEquality,
universeIsType,
inhabitedIsType,
Error :memTop,
lambdaEquality_alt,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename
Latex:
\mforall{}C:SmallCategory. \mforall{}X,Delta:ps\_context\{j:l\}(C). \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}.
\mforall{}s:psc\_map\{j:l\}(C; Delta; X). \mforall{}I:cat-ob(C). \mforall{}a:Delta(I).
(presheaf-pi-family(C; X; A; B; I; (s)a) = presheaf-pi-family(C; Delta; (A)s; (B)(s o p;q); I; a))
Date html generated:
2020_05_20-PM-01_28_55
Last ObjectModification:
2020_04_02-PM-01_56_55
Theory : presheaf!models!of!type!theory
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