Nuprl Lemma : csm-subset-subtype
∀[A,B,Y,Z:j⊢]. (Y j⟶ A ⊆r Z j⟶ B) supposing (sub_cubical_set{j:l}(A; B) and sub_cubical_set{j:l}(Z; Y))
Proof
Definitions occuring in Statement :
sub_cubical_set: Y ⊆ X,
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
sub_cubical_set: Y ⊆ X,
sub_ps_context: Y ⊆ X,
cube_set_map: A ⟶ B,
csm-id: 1(X),
pscm-id: 1(X)
Lemmas referenced :
pscm-subset-subtype,
cube-cat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
sqequalRule
Latex:
\mforall{}[A,B,Y,Z:j\mvdash{}].
(Y j{}\mrightarrow{} A \msubseteq{}r Z j{}\mrightarrow{} B) supposing (sub\_cubical\_set\{j:l\}(A; B) and sub\_cubical\_set\{j:l\}(Z; Y))
Date html generated:
2020_05_20-PM-02_34_12
Last ObjectModification:
2020_04_03-PM-08_44_38
Theory : cubical!type!theory
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