### Nuprl Lemma : presheaf-subset_wf

`∀[C:SmallCategory]. ∀[F:Presheaf(C)]. ∀[P:I:cat-ob(C) ⟶ (ob(F) I) ⟶ ℙ].`
`  F|I,rho.P[I;rho] ∈ Presheaf(C) supposing stable-element-predicate(C;F;I,rho.P[I;rho])`

Proof

Definitions occuring in Statement :  presheaf-subset: `F|I,rho.P[I; rho]` stable-element-predicate: `stable-element-predicate(C;F;I,rho.P[I; rho])` presheaf: `Presheaf(C)` functor-ob: `ob(F)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` member: `t ∈ T` apply: `f a` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` presheaf-subset: `F|I,rho.P[I; rho]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` presheaf: `Presheaf(C)` all: `∀x:A. B[x]` so_apply: `x[s1;s2]` prop: `ℙ` so_apply: `x[s]` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` cat-ob: `cat-ob(C)` pi1: `fst(t)` type-cat: `TypeCat` so_apply: `x[s1;s2;s3;s4]` so_lambda: `λ2x y.t[x; y]` guard: `{T}` stable-element-predicate: `stable-element-predicate(C;F;I,rho.P[I; rho])` implies: `P `` Q` top: `Top` functor-arrow: `arrow(F)` pi2: `snd(t)` compose: `f o g`
Lemmas referenced :  mk-presheaf_wf functor-ob_wf op-cat_wf small-category-subtype type-cat_wf subtype_rel-equal cat-ob_wf cat_ob_op_lemma subtype_rel_self cat-arrow_wf set_wf stable-element-predicate_wf presheaf_wf small-category_wf functor-arrow-id op-cat-id cat_arrow_triple_lemma cat_id_tuple_lemma cat-comp_wf functor-arrow-comp op-cat-arrow op-cat-comp cat_comp_tuple_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache lambdaEquality setEquality applyEquality instantiate hypothesisEquality hypothesis independent_isectElimination dependent_functionElimination functionExtensionality universeEquality lambdaFormation setElimination rename dependent_set_memberEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality functionEquality cumulativity independent_functionElimination voidElimination voidEquality applyLambdaEquality

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[F:Presheaf(C)].  \mforall{}[P:I:cat-ob(C)  {}\mrightarrow{}  (ob(F)  I)  {}\mrightarrow{}  \mBbbP{}].
F|I,rho.P[I;rho]  \mmember{}  Presheaf(C)  supposing  stable-element-predicate(C;F;I,rho.P[I;rho])

Date html generated: 2017_10_05-AM-00_51_02
Last ObjectModification: 2017_10_03-PM-03_19_41

Theory : small!categories

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