Nuprl Lemma : psc-predicate_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[P:I:cat-ob(C) ⟶ X(I) ⟶ ℙ{[i | j']}].
(psc-predicate(C; X; I,rho.P[I;rho]) ∈ ℙ{[i | j']})
Proof
Definitions occuring in Statement :
psc-predicate: psc-predicate(C; X; I,rho.P[I; rho]),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
member: t ∈ T,
function: x:A ⟶ B[x],
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
psc-predicate: psc-predicate(C; X; I,rho.P[I; rho]),
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
stable-element-predicate: stable-element-predicate(C;F;I,rho.P[I; rho]),
all: ∀x:A. B[x],
ps_context: __⊢,
cat-functor: Functor(C1;C2),
and: P ∧ Q,
uimplies: b supposing a,
type-cat: TypeCat,
implies: P ⇒ Q,
so_apply: x[s1;s2],
I_set: A(I)
Lemmas referenced :
cat-ob_wf,
I_set_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
cat-arrow_wf,
I_set_pair_redex_lemma,
ob_pair_lemma,
subtype_rel-equal,
op-cat_wf,
cat_ob_op_lemma,
cat_ob_pair_lemma,
functor-arrow_wf,
type-cat_wf,
op-cat-arrow
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
universeEquality,
instantiate,
applyEquality,
sqequalRule,
functionEquality,
cumulativity,
because_Cache,
setElimination,
rename,
productElimination,
dependent_functionElimination,
Error :memTop,
independent_isectElimination,
lambdaEquality_alt
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[P:I:cat-ob(C) {}\mrightarrow{} X(I) {}\mrightarrow{} \mBbbP{}\{[i | j']\}].
(psc-predicate(C; X; I,rho.P[I;rho]) \mmember{} \mBbbP{}\{[i | j']\})
Date html generated:
2020_05_20-PM-01_23_25
Last ObjectModification:
2020_04_02-AM-11_56_58
Theory : presheaf!models!of!type!theory
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