Nuprl Lemma : pscm-equal2
∀[C:SmallCategory]. ∀[A,B:ps_context{j:l}(C)]. ∀[f,g:psc_map{j:l}(C; A; B)].
f = g ∈ psc_map{j:l}(C; A; B) supposing ∀K:cat-ob(C). ∀x:A(K). ((f K x) = (g K x) ∈ B(K))
Proof
Definitions occuring in Statement :
psc_map: A ⟶ B,
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
ps_context: __⊢,
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),
cat-ob: cat-ob(C),
guard: {T}
Lemmas referenced :
pscm-equal,
subtype_rel_dep_function,
cat-ob_wf,
op-cat_wf,
cat-arrow_wf,
type-cat_wf,
functor-ob_wf,
I_set_wf,
subtype_rel-equal,
cat_ob_op_lemma,
subtype_rel_self,
small-category-cumulativity-2,
psc_map_wf,
ps_context_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
applyEquality,
instantiate,
cumulativity,
hypothesis,
sqequalRule,
lambdaEquality_alt,
because_Cache,
universeIsType,
functionEquality,
independent_isectElimination,
dependent_functionElimination,
lambdaFormation_alt,
functionExtensionality_alt,
functionIsType,
equalityIstype,
universeEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:ps\_context\{j:l\}(C)]. \mforall{}[f,g:psc\_map\{j:l\}(C; A; B)].
f = g supposing \mforall{}K:cat-ob(C). \mforall{}x:A(K). ((f K x) = (g K x))
Date html generated:
2020_05_20-PM-01_23_59
Last ObjectModification:
2020_04_01-AM-10_47_10
Theory : presheaf!models!of!type!theory
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