Nuprl Lemma : psdcff-inj-injection
∀[C:SmallCategory]. ∀[A,B:Type]. ∀[X:ps_context{j:l}(C)]. ∀[I:cat-ob(C)]. ∀[a:X(I)].
Inj(presheaf-fun-family(C; X; discr(A); discr(B); I; a);A ⟶ B;λw.psdcff-inj(I;w))
Proof
Definitions occuring in Statement :
psdcff-inj: psdcff-inj(I;w),
discrete-presheaf-type: discr(T),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
I_set: A(I),
ps_context: __⊢,
inject: Inj(A;B;f),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
psdcff-inj: psdcff-inj(I;w),
discrete-presheaf-type: discr(T),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
presheaf-type-at: A(a),
pi1: fst(t),
prop: ℙ,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
presheaf-type-at_wf,
discrete-presheaf-type_wf,
psc-restriction_wf,
cat-arrow_wf,
presheaf-type-ap-morph_wf,
cat-comp_wf,
subtype_rel-equal,
psdcff-inj_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-fun-family_wf,
I_set_wf,
cat-ob_wf,
ps_context_wf,
istype-universe,
small-category_wf,
subtype_rel_self,
equal_wf,
squash_wf,
true_wf,
cat-id_wf,
cat-comp-ident2,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
functionExtensionality,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
functionIsType,
universeIsType,
equalityIstype,
independent_isectElimination,
instantiate,
cumulativity,
inhabitedIsType,
lambdaEquality_alt,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:Type]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[I:cat-ob(C)]. \mforall{}[a:X(I)].
Inj(presheaf-fun-family(C; X; discr(A); discr(B); I; a);A {}\mrightarrow{} B;\mlambda{}w.psdcff-inj(I;w))
Date html generated:
2020_05_20-PM-01_35_53
Last ObjectModification:
2020_04_02-PM-06_35_57
Theory : presheaf!models!of!type!theory
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