Nuprl Lemma : psdcff-inj-bijection
∀[C:SmallCategory]. ∀[A,B:Type]. ∀[X:ps_context{j:l}(C)]. ∀[I:cat-ob(C)]. ∀[a:X(I)].
Bij(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: __⊢,
biject: Bij(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],
biject: Bij(A;B;f),
and: P ∧ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
surject: Surj(A;B;f),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
psdcff-inj: psdcff-inj(I;w),
discrete-presheaf-type: discr(T),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a)
Lemmas referenced :
psdcff-inj-injection,
small-category-cumulativity-2,
ps_context_cumulativity2,
I_set_wf,
cat-ob_wf,
ps_context_wf,
istype-universe,
small-category_wf,
psdcff-inj_wf,
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
cat-arrow_wf,
istype-top,
cat-comp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
cumulativity,
universeIsType,
inhabitedIsType,
universeEquality,
lambdaFormation_alt,
dependent_pairFormation_alt,
equalityIstype,
because_Cache,
functionIsType,
dependent_functionElimination,
Error :memTop,
dependent_set_memberEquality_alt,
lambdaEquality_alt
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:Type]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[I:cat-ob(C)]. \mforall{}[a:X(I)].
Bij(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_57
Last ObjectModification:
2020_04_02-PM-06_36_09
Theory : presheaf!models!of!type!theory
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