Nuprl Lemma : presheaf-fun-comp_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A,B,E:{X ⊢ _}]. ∀[g:{X ⊢ _:(A ⟶ B)}]. ∀[f:{X ⊢ _:(B ⟶ E)}].
((f o g) ∈ {X ⊢ _:(A ⟶ E)})
Proof
Definitions occuring in Statement :
presheaf-fun-comp: (f o g),
presheaf-fun: (A ⟶ B),
presheaf-term: {X ⊢ _:A},
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-fun-comp: (f o g),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T
Lemmas referenced :
presheaf-lam_wf,
presheaf-app_wf_fun,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-ap-type_wf,
psc-fst_wf,
pscm-ap-term_wf,
presheaf-fun_wf,
subtype_rel-equal,
presheaf-term_wf,
psc-snd_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
hypothesis,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A,B,E:\{X \mvdash{} \_\}]. \mforall{}[g:\{X \mvdash{} \_:(A {}\mrightarrow{} B)\}].
\mforall{}[f:\{X \mvdash{} \_:(B {}\mrightarrow{} E)\}].
((f o g) \mmember{} \{X \mvdash{} \_:(A {}\mrightarrow{} E)\})
Date html generated:
2020_05_20-PM-01_31_19
Last ObjectModification:
2020_04_02-PM-05_52_05
Theory : presheaf!models!of!type!theory
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