### Nuprl Lemma : presheaf-subset-and

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

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

Date html generated: 2017_10_05-AM-00_51_10
Last ObjectModification: 2017_10_03-PM-03_30_44

Theory : small!categories

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