Nuprl Lemma : pscm-equal
∀[C:SmallCategory]. ∀[A,B:ps_context{j:l}(C)]. ∀[f:psc_map{j:l}(C; A; B)]. ∀[g:I:cat-ob(C) ⟶ A(I) ⟶ B(I)].
f = g ∈ psc_map{j:l}(C; A; B) supposing f = g ∈ (I:cat-ob(C) ⟶ A(I) ⟶ B(I))
Proof
Definitions occuring in Statement :
psc_map: A ⟶ B,
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
psc_map: A ⟶ B,
member: t ∈ T,
subtype_rel: A ⊆r B,
ps_context: __⊢,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat,
I_set: A(I),
guard: {T},
implies: P ⇒ Q,
nat-trans: nat-trans(C;D;F;G),
prop: ℙ
Lemmas referenced :
nat-trans-equal,
op-cat_wf,
type-cat_wf,
small-category-cumulativity-2,
subtype_rel_dep_function,
cat-ob_wf,
I_set_wf,
cat-arrow_wf,
functor-ob_wf,
subtype_rel-equal,
cat_ob_op_lemma,
subtype_rel_self,
equal_functionality_wrt_subtype_rel2,
subtype_rel_set,
equal_wf,
cat-comp_wf,
functor-arrow_wf,
psc_map_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
cumulativity,
lambdaEquality_alt,
functionEquality,
universeIsType,
independent_isectElimination,
dependent_functionElimination,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
equalityIstype,
functionIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:ps\_context\{j:l\}(C)]. \mforall{}[f:psc\_map\{j:l\}(C; A; B)]. \mforall{}[g:I:cat-ob(C)
{}\mrightarrow{} A(I)
{}\mrightarrow{} B(I)].
f = g supposing f = g
Date html generated:
2020_05_20-PM-01_23_57
Last ObjectModification:
2020_04_01-AM-09_57_00
Theory : presheaf!models!of!type!theory
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